Ind. Eng. Chem. Res. 1990,29, 2163-2166
2163
and a free chlorine source simultaneously.
Sunlight Exposure (hr)
Figure 3. Comparison of the stabilities of DC and free chlorine at pH 7.0, 22-24 OC, in demand-free water in the presence of direct sunlight.
the contact times employed in these experiments. DC gave a 6 log reduction of P. aeruginosa at 5 mg/L total chlorine within 5 min at pH 7.0, 22 "C, DFW. A t heavy demand (SDW, pH 9.5, 4 "C), a 6 log reduction of S. Aureus was obtained in less than 4 h for a beginning concentration of 10 mg/L total chlorine from DC. It is well-known that S. aureus is a very resistant organism to disinfection by chlorination. Prior data for other N-halamine compounds developed in these laboratories, as well as for free chlorine, under the same experimental conditions as employed in this work have been reviewed recently (Worley and Williams, 1988). The results obtained for algae were impressive. As long as any measurable DC remained in solution, the algal concentration (asdetermined by absorption measurement at 750 nm) did not increase from zero. The experiments were run for sufficient time (over 20 days) such that the control aquarium showed heavy algal growth. It was also found that the algal concentration in an aquarium already containing heavy growth was decreased by 77% within 24 h upon addition of 10 mg/L total chlorine provided by DC. Thus, it would appear that DC will be useful for preventing algal growth in swimming pools and other large water sources. Finally, recent experiments in these laboratories have shown that DC can be formed rapidly (a few seconds) in situ if a free chlorine source such as Ca(OCl), is added to a water containing 2,2,5,5-tetramethylimidazolidin-4-one, precursor to laboratory synthesis of DC. Therefore, a possible means of disinfecting a large body of water such as a swimming pool would be the addition of the precursor
Conclusions The new N-halamine compound, 1,3-dichloro-2,2,5,5tetramethylimidazolidin-4-one(DC), which can be prepared inexpensively from the readily available materials acetone, NaCN, NH4C1, (NH4).$, and chlorine, is a stable compound in water that is effective as a biocide for vegitative bacteria and algae. It has not yet been tested against viruses,protozoa, and fungi. It should be especially useful for long-term disinfection of aqueous systems such as swimming pools, hot tubs, and cooling towers but could also have application for hard surfaces. Further testing is in progress, as well as the syntheses of derivatives of DC and brominated analogues. The results of these studies will be reported in due course. Registry No. DC, 128780-87-0. Literature Cited American Public Health Association. Standard Methods for the Examination of Water and Wastewater, 16th ed.; APHA Washington, DC, 1985; pp 306-309,950-954. Barnela, S. B.; Worley, S. D.; Williams, D. E. Syntheses and Antibacterial Activity of New N-Halamine Compounds. J . Pharm. Sci. 1987, 76, 245-247. Christian, J. D. 4-Imidazolidinethiones. J . Org. Chem. 1957, 22, 396-399. Nelson, G . D. Chloramines and Bromamines. in Kirk-Othmer Encyclopedia of Chemical Technology, 3rd ed.; Wiley Interscience: New York, 1979; p 565. Starr, R. C. The Culture Collection of Algae at the University of Texas. J. Phycol. 1978, 14, 47-100. Toda, T.; Mori, E.; Horiuchi, H.; Murayama, K. Studies on Stable Free Radicals. X. Photolysis of Hindered N-Chloramines. Bull. Chem. SOC.Jpn. 1972,45, 1802-1806. Williams, D. E.; Worley, S. D.; Barnela, S. B.; Swango, L. J. The Bacterial Activities of Selected Organic N-Halamines. Appl. Environ. Microbiol. 1987, 53, 2082-2089. Worley, S. D.; Williams, D. E. Halamine Water Disinfectants. CRC Crit. Rev. Environ. Control 1988, 18, 133-175. Worley, S. D.; Williams, D. E.; Barnela, S. B. The Stabilities of New N-Halamine Water Disinfectants. Water. Res. 1987,21,983-988.
Te-Chen Tsao, Delbert E. Williams, S. Davis Worley* Department of Chemistry Auburn University Auburn University, Alabama 36849 Received for review February 14, 1990 Accepted July 10, 1990
When To Use Cascade Control A comparative study of cascade versus feedback control of single-input, single-output (SISO) systems has been made. The primary and secondary processes are represented by first-order with dead-time models. The results show the relative benefits due to cascade control vis-&vis feedback control for varying process parameters. They can be used to decide when to specify cascade control. Finally, tuning charts are presented for designing the primary controller of the cascade control system. It is believed that the material in the paper will be useful to those who are interested in the design and implementation of cascade control systems. Introduction Cascade control is used to improve the dynamic response of a feedback control loop to disturbances in the manipulated variable. It is important for the designers to know when to specify cascade control and what improvements can be expected, say, in terms of ITAE (integral of time 0888-5885/90/2629-2163$02.50/0
X absolute error), vis-&vis feedback control, since the choice of cascade control requires an additional measurement device/transmitter and an additional controller. Information in the published literature on the relative benefits of cascade control appears to be rather limited. Among the relevant works is that of Harriott (1964), who
0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
2164
i
I
I
i
Figure 1. Block diagram of cascade control system.
provided some insights into the basic characteristics of cascade control. Frank and Worley (1956) documented the relative improvement in the performance due to cascade control in comparison with feedback control of systems having two first-order lags representing the primary process and three first-order lags representing the secondary process. McMillan (1982) demonstrated the benefits of cascade control for typical self-regulating, integrating, and runaway processes. The foregoing studies are of limited use since they do not provide quantitative information on the improvement in performance from cascade control for user-selected models containing dead time. Schork and Deshpande (1978) applied double cascade control to an industrial polymer film drying operation. Krishnaswamy and Rangaiah (1987) showed how the dynamics of both the primary and the secondary processes can be identified with a single pulse test. Deshpande (1980) studied cascade control applied to an industrial polymerization reactor and showed how one could identify the dynamics of an open-loop unstable process by the pulse testing method. It is widely accepted that complex dynamic processes can often be represented by a first-order with dead-time (FODT) model. If the primary and secondary processes are represented by an FODT model, then one needs to know, for the particular choice of model parameters, whether to specify cascade control and, if so, what would be the extent of improvement in comparison with feedback control. In this study, extensive simulation work has been done, assuming this type of process representation, and general guidelines have been developed to help answer these questions. Simulation Study A block diagram of the cascade control system studied is shown in Figure 1. A PI controller serves as the primary controller, and a proportional controller is specified as the secondary controller. The PI/P combination is commonly used in industry since it gives good performance and contains only three tuning parameters; the integral action in the primary controller is required to ensure offset-free performance of the primary process output. The secondary controller was tuned according to the equation (Lopez, 1967) -1.085 KIKcl = O."897(
$)
To tune the primary controller, the dynamics of the open outer loop containing the closed inner loop was approximated by a second-order plus dead-time model (Sundaresan et al., 1978) and the tuning charts developed by Lopez (1967) were used. ITAE was used as the optimization criterion in both instances. The corresponding single-loop P I controller was also tuned according to the ITAE settings given by Lopez (1967). The closed-loop responses of the cascade and feedback systems to a step change in L1 and L2 were evaluated by CSMP I11 (Continuous System Modeling Program) on an IBM mainframe computer system. The closed-loop response equations are presented in the Appendix. The dead
- 10
02
06
01
08
10
RIR
Figure 2. ITAE ratio for secondary loop disturbance.
time to time constant ratios (R, = 8 J T 1 and R2 = e2/T2) of 0.1,0.3,0.6, and 1.0 were considered for each (primary and secondary) process. In addition, two discrete values of 0.2 and 1.0 were chosen for the secondary to primary time constant ratio, R = T l / T 2 These values generated a total of 32 simulation conditions having small or large dead times in either or both primary and secondary processes and different loop dynamics (i.e., outer loop slower than or as fast as the inner loop). Results and Discussion The real benefit of cascade control can be seen when disturbances enter the secondary loop at L1.For all 32 simulated processes, the ratios of ITAEgenerated by single loop PI control to those generated by cascade PI/P control for a unit step change in L1 have been analyzed. These ratios essentially represent, in terms of the chosen criterion, the relative superiority of cascade control over simple feedback control. The results obtained are shown in Figure 2 as plots of ITAE ratio versus RIR with R2 as an additional parameter. It may be noted that, with reference to the major process time constant T2,RIR and R2 represent, respectively, the variations in secondary and primary process delays. The values of R chosen are such that they provide primary time constants that are greater than or equal to the secondary process time constant. It may be seen from Figure 2 that irrespective of the values assigned to R and R2,cascade control shows a steep improvement in performance as the inner loop dead time (R,R) decreases. Thus, for R = 1, while cascade control performs about 2 times better than ordinary feedback when R1 = 1, its performance shows a 15-50-fold improvement when R1is reduced to 0.1. This is because as is reduced, relative to T I ,the critical frequency of the inner loop increased, resulting in a larger gain of the secondary controller. The results further reveal that, in contrast to the change in R1, a change in the outer loop dead time (R,) does not have such a pronounced effect on the relative control performance. The effect of R2 on the ITAE ratio is marginal when R1 and R are close to l . When either of these parameters decreases, the ITAE ratio begins to vary with R2. Thus, for R = 1 and Rl = 0.1, while cascade control
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2165
_.
0
io
5
15
TIME Figure 3. Control system responses for secondary load. lo t R,
I
c
2
'0
001
008
012
016
020
w
LR1T
Figure 4. ITAE ratio for primary loop disturbance.
performs about 15 times better than feedback control when
R2 = 1, its performance shows about 50-fold improvement
when R2 is reduced to 0.1. A performance comparison between cascade P I / P and feedback PI in terms of actual responses is provided in Figure 3 for a unit step change in L1. These curves for R1 = 0.1, R2 = 0.1, and R = 1 are illustrative of some typical results. For this particular case, the ITAE ratio is more than 50. Obviously, the performance of cascade control is significantly better than that of feedback. Another interesting point to note is that the response curve in the case of cascade control is almost free from oscillations, while in the case of ordinary feedback, oscillations are predominant. During normal operation, the disturbances may also occur in the primary loop; hence, it is important that the cascade control system performs at least as good as ordinary feedback control for changes in LP All the process situations considered earlier were therefore simulated again with the same ITAE controller settings employed, and the responses to a unit step change in L2 were obtained both for single loop and cascade control. As before, the results were characterized in terms of ITAE ratios as shown in Figure 4. Figure 4 reveals that for all cases considered, the ITAE ratio is greater than 1.0, implying that the performance of cascade control is better than or as good as feedback control. The ITAE ratio generally increases with a decrease in dead time in either the secondary or the primary loop. Actually, cascade loop performs 3-10 times better than the feedback loop when dead-time ratios (R, and R2) are small.
-0.1 1 0
2
1
6
B
10
12
TIME Figure 5. Control system responses for primary load.
The responses of cascade P I / P and feedback PI for primary load disturbances are presented in Figure 5 for R1 = 0.1, R2 = 0.1, and R = 1. The improvement in the performance of the cascade loop corresponds to an ITAE ratio of nearly 10. Further, as in the case of secondary load upsets (Figure 3), oscillations in the PI/P response are not large or numerous. In contrast, the response with singleloop control shows significant oscillations. Finally, a word of caution about a cascade loop with a small dead time in the inner loop that might tempt the designer to tweak up the gain of the secondary controller. A large number of secondary loops used in industry are flow loops where their function is to guard against exogenous disturbances such as pressure variations upstream of the flow control valve. In these situations, a larger inner loop gain on a proportional-type controller will be stable and very responsive. However, flow loops also generally experience an appreciable amount of sensor noise from fluid turbulence. A more conservative, lower gain of the secondary controller may provide nearly the same level of disturbance rejection in the inner loop while saving a great deal of wear and tear on the control valve packing resulting from an excessively nervous secondary controller gain. Tuning Relations. Several tuning relations that correlate PID-type controller parameters with process parameters (dead time and time constant) are readily available (e.g., Smith and Corripio, 1985) for single-loop control either in graphical form or as equations. These relations also aid in tuning the secondary controller of cascade systems. However, no correlations have been reported in the open literature for tuning the primary controller, although they would be of practical use. Design charts that predict the primary controller (PI) settings for minimizing ITAE due to secondary load disturbances have been developed, and two are shown in Figure 6. The plots are valid for primary and secondary processes represented by first-order plus time-delay models with the secondary loop closed by using a P controller. Conclusions A comparative study of cascade versus feedback control for first-order with dead-time processes has been carried out. On the basis of simulation results, charts have been prepared that show the extent of improvement possible with cascade control. Design charts for finding the tuning constants of the primary controller have been presented. The material in this paper will help the designer decide whether to specify cascade control for SISO systems and if so how to tune its primary controller.
2166 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
C,(s) = (K1K2e-(81+e2)s/(T1s + l)(T,s + 1))/
{ s [ l+ KclKle-B~s/(Tls + 1 ) + Kc1K1Kc2K2(1 + 1 / T i 2 ~ ) e - ( 8 ~ + 8 2+ ) sl)(T,s / ( T l ~ + l ) ] ]= [K1K2e-(81+82)S]/(~[(T1~ + 1)(T2s+ 1 ) + KclKle-B1S(T2s + 1 ) + Kc,KIK,,K,(l + l/Ti2s)e-(81+82)s]] (Al)
Introducing the transformation s‘ = sT2 into the above equation, which is equivalent to introducing a dimensionless time defined by t’ = t / T 2 ,the normalized time domain system response could be described by 1 C,(t’) = Y,(t’) =
K, K2
+
L-1{e-(RIR+R2)S‘/s’[(s’l)(Rs’+ 1) + K’,,(s’ + 1)emRIRs’+ K’clK ’c2( 1 + 1/ T:,s ’) e-(RIR+Rl)s‘](A2)
It may be noted that in the above expression the dynamic characteristics of the response are expressed in terms of dimensionless process parameters R1 = 01/T1; R2 = OZ/Tz; R = T1/T2 (A3) r:
,
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Figure 6. Primary controller settings for minimizing ITAE due to secondary loop disturbances.
Nomenclature C = system response K = process steady-state gain K, = proportional controller gain L = disturbance input R = Ti/T, R1 = W T 1 R2 = 02IT2 s = Laplace transform operator s’= S T ~ t = time t’ = dimensionless time, t / T 2 T = process time constant Td = derivative time Ti = integral time Y = normalized system response Greek L e t t e r 0 = process time delay Subscripts 1 = inner loop 2 = outer loop Superscripts
’ = dimensionless quantities (see eq A4)
Appendix Referring to Figure 1, response C2 of the control system for a unit step change in disturbance signal L , could be written as
and controller parameters K$1 = KIKcl; K:. = KZK,,;
T’i, = Ti,/T2 (A4)
Literature Cited Deshpande, P. B. Process Identification of Open-loop Unstable Systems. AZChE J . 1980,26, 305-308. Franks, R. G.; Worley, C. W. Quantitative Analysis of Cascade Control. Ind. Eng. Chem. 1956,48, 1074-1079. Harriott, P. Process Control; McGraw Hill: New York, 1964. Krishnaswamy, P. R.; Rangaiah, G. P. Tuning of Cascade Control Loops. Proceedings of the ACC, Minneapolis, 1987; pp 1158-1160. Lopez, A. M. PhD. Thesis, Louisiana State University, Baton Rouge, 1967. McMillan, G. K. Effect of Cascade Control on Loop Performance. Proceedings of the ACC, Arlington, 1982; pp 363-368. Schork, F. J.; Deshpande, P. B. Double Cascade Controller Tested. Hydrocarbon Process. 1978, 57 (7), 113-117. Smith, C. A.; Corripio, A. B. Principles and Practice of Automatic Process Control; John Wiley: New York, 1985. Sundaresan, K. R.; Chandraprasad, C.; Krishnaswamy, P. R. Evaluating Parameters from Process Transients. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 237-241.
* Author to whom correspondence should be addressed. Peruvemba R. Krishnaswamy,* Gade P. Rangaiah Department of Chemical Engineering National University of Singapore Singapore 051I, Singapore Radha K. Jha Department of Chemical Engineering Banaras Hindu University Varanasi 221005, India Pradeep B. Deshpande Department of Chemical Engineering University of Louisville Louisville, Kentucky 40292 Received for review March 6, 1990 Accepted June 25,1990
