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206
Table IV. Maximum C O Product Recovery at Different CO Product Purities a n d Optimum Input Variables max CO recovery, CO purity, % % VI v2 v3 -1.215 -0.524 1.215 80.0 99.999 0.171 -1.215 -0.850 85.0 99.411 90.0 99.215 0.200 x -0.043 -1.215 -0.557 -1.215 95.0 95.980 1.215 -1.086 -1.215 87.688 1.215 98.0
Maximize CO Recovery for a Given CO Purity. The objective function is F ( C 0 recovery, 70)= -[57.122 + 2.490V1 + 10.130V2 25.090v3 - 0.O526vI2 - 2.747V2 + 8.216V32 0.763V1V2 - 1.612V,V3 - 0.247V3V1] (17) with inequality constraints eq 7-9 and equality constraints CO purity, % = [rhs expression from eq 161 = constant Table IV shows the maximum CO recovery for different values of CO purities and optimum conditions. The CO recovery decreases with an increase in CO purity. The foregoing demonstrates that for a given separation requirement, the optimization technique can be used efficiently to locate the optimum PSA cycle conditions. The technique can be expanded to optimize more than three input variables and can be used for other PSA cycles. Moreover, total cost can also be optimized by expressing
it as a function of the input variables.
Acknowledgment This work was supported by the U S . Department of Energy under Grant DE-AC21-85MC22060. Literature Cited Afimiwala, K. A. “Computer Programs for Optimization Including Applications”, M.S. Thesis, SUNY at Buffalo, Buffalo NY, 1973. Afimiwala, K. A.; Mayne, R. W. J. Eng. Ind. ASME 1974, 96(1), 1. Beveridge, G. S. G.; Schechter, R. S. Optimization: Theory and Practice; McGraw-Hill: New York, 1970; Chapter 3, p 41. Chihara, K.; Kondo, A. Paper presented at the Engineering Foundation Conference on Adsorption, Santa Barbara, CA, May 1986; to be published by Engineering Foundation, New York. Doshi, K. J.; Katira, C. H.; Stewart, H. A. AZChE Symp. Ser. 1971, 67 (1171, 90. Kapoor, A., SUNY at Buffalo, Buffalo, NY, unpublished results, 1987. Kapoor, A.; Yang, R. T. Sep. Sci. Technol. 1987, in press. Reklaitis, G. V.; Ravidran, A.; Ragsdell, K. M. Engineering Optimization: Methods and Applicacions; Wiley: New York, 1983; Chapter 6, p 216. Yang, R. T . Gas Separation by Adsorption Processes; Butterworth: Boston, 1987; Chapter 7, p 237.
A. Kapoor, R. T. Yang* Department of Chemical Engineering S t a t e University of New York a t Buffalo Buffalo, New York 14260 Received for review February 2, 1987 Revised manuscript received September 29, 1987 Accepted October 16, 1987
The Concept of “Eigenstructure”in Process Control Much of t h e work in multivariable process control has been directed at finding control structures that minimize interaction among loops and decouple the system. This paper claims that this approach is flawed. What is really important in the vast majority of chemical and petroleum processes is a structure t h a t does t h e best job in rejecting load disturbances. This inherent or intrinsic “eigenstructure” (choice of controlled and manipulated variables and their pairing) is that configuration which yields a system that is n a t u r d y self-regulating for load disturbances and self-optimizing. Eigenstructure is a unifying concept that links several previously published approaches to the process control problem. The literature in the area of multivariable control is quite extensive. Much of it has been directed at the problem of control system structure. The structure must be specified before the subsequent steps of controller tuning and hardware implementation can proceed. Therefore, the initial problem in process control is structure. Questions to be answered are as follows: (1)What variables should be controlled? (2) What variables should be manipulated? This means not only what control valves or flow rates to manipulate, but also consideration of the possibility of using flow ratios, sums or differences of flow rates, heat removal or addition rates, etc. (3) How should these controlled and manipulated variables be linked together? Should we use a diagonal (multiloop SISO) controller or a full multivariable controller? Many workers have concentrated on choosing the structure such that interaction among the control loops is minimized. The use of decouplers assumes coupling is undesirable. The much-used (and abused) Relative Gain Array (RGA) method and the Inverse Nyquist Array (INA) method are based on the assumption that control loop interaction is bad. This may be true in systems where 0888-5885/88/2627-0206$01.50/0
set-point changes are the principal disturbance. But set-point disturbances are usually much less frequent in chemical process control than load disturbances. Most industrial applications require a control system that can hold the process at desired values of performance (composition, yield, etc.) in the face of load disturbances such as variations in feed composition and throughput. In fact, as Niederlinski (1971) pointed out over a decade ago, designing control systems such that they are noninteracting can degrade the performance of the system in rejecting load disturbances. The predictive methods (such as DMC and IMC) assume that the controlled and manipulated variables have already been chosen and that a multivariable controller is desired. Thus these methods are essentially tuning procedures. The structure is specified, and the job is simply to calculate controller tuning parameters that give a reasonable compromise between performance and robustness. The purpose of this paper is to put forward the notion that each process has an intrinsically self-regulating control structure which makes the system as insensitive as possible to load disturbances and is self-optimizing. Thus a control system design philosophy is proposed that contains as its 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 207 first step the discovery of this eigenstructure. Waller et al. (1986) have recently discussed a similar approach. This paper also makes the claim that the eigenstructure concept provides a common thread that appears in the papers of several workers in the process control area: (1) Buckley (1964) with his overall plant control strategy; (2) Luyben (1975) with his impact of steady-state energy consumption on control system structure; (3) Douglas (Fisher et al., 1984) with his steady-state plant-wide control ideas; (4) McAvoy (Stanley et al., 1985) with his Relative Disturbance Gain; (5) Tyreus (1987) with his integration of steady-state optimization into the regulatory control structure; (6) Georgakis (1986) with his reaction rate or extensive variable control. We will expand on each of these in the following discussion.
Relationships of Previous Work with the Eigenstructure Concept Buckley (1964) approached the problem of plant-wide control by splitting the problem horizontally, i.e., considering the slow material-balance control structure of the entire plant first and then later establishing the faster composition control structure of each individual unit. This two-level approach is in contrast to the vertical splitting of the plant-wide control problem that most academics attempt to employ, i.e., slicing the plant up into many little subunits in series. Buckley’s material-balance control structure (slow liquid level and gas pressure loops) produced a plant that intrinsically handled disturbances well. The effects of load changes were attenuated as they worked their way through the process. This slow material-balance control is a component of eigenstructure. Luyben (1975) pointed out that the optimum control structure that minimized energy consumption in distillation columns required controlling product compositions at both ends of the column (dual composition control). However he suggested that a more simple, single-end control structure could often be found that used very little additional energy. Steady-state rating programs were used to calculate how the manipulated variables (vapor boilup, reflux, and reflux ratio) had to change a t steady state as feed composition changed over the expected range of variability to keep product compositions constant at their specified values. If one of these manipulated variables was found to vary only slightly, a simple control structure was recommended hold this variable constant at its maximum value (found over the range of feed compositions) and manipulate the other variable to control the composition of only one product. Since excess reflux was used, the composition of the uncontrolled product would always be at or above specification. Thus the complexities and potential interaction problems of dual composition control could be avoided with very little energy penalty. Feed rate disturbances could be effectively handled by ratio schemes in a feed-forward sense. This is one of the first examples of finding an eigenstructure that yields a self-optimizing and more stable control system. This same basic notion was extended to an entire chemical plant by Douglas. The paper by Fisher et al. (1984) proposed the use of steady-state rating programs to find the optimum operating conditions of the overall plant for different disturbances. Then simple relationships were found between controlled and manipulated variables such that the system was inherently held a t or near the optimum point. The specific example that Douglas used to illustrate the concepts is the hydrodealkylation of toluene (HDA process). The plant consists of a preheater, furnace, reactor, cooler, flash drum, recycle gas compressor, and three distillation columns. Given a toluene feed rate,
there are 22 manipulated variables: hydrogen feed, furnace fuel, quench flow to the reactor effluent, cooling water to the partial condenser, purge gas flow, recycle gas flow, liquid from the flash drum, and five manipulated variables in each column (two products, coolant, reflux, and reboiler heat input). Of course, 11 of these are set by inventory loops (pressure and liquid level in the flash drum and two liquid levels and a pressure in each distillation column). The remaining 11manipulated variables should, in theory, be established by an optimization procedure for each load condition. Changes in production rate, cooling water inlet temperature, and hydrogen feed gas purity were considered. However, Douglas showed that a more simple, suboptimum control structure held the plant quite close to the optimum. Recycle gas flow was maximized. Three variables were held constant: toluene recycle flow, vapor rate in the stabilizer column, and reflux flow in the recycle column. Actually, it was found that no reflux should be used in the recycle column, making it just a simple stripper. The remaining seven manipulated variables were used to control (1)hydrogen-to-aromatics ratio at the reactor inlet, (2) exit temperature of the cooling water from the partial condenser, (3) quench temperature, (4) benzene product composition, (5) bottoms composition in the product column, (6) bottoms composition in the recycle column, and (7) distillate composition in the stabilizer column. Although no consideration was given to dynamic effects on the selection of the pairing, the concept fits into the framework of eigenstructure. McAvoy, recognizing that control system design involves more than just the RGA, proposed the use of the Relative Disturbance Gain. Stanley et al. (1985) presented this index which looks at the changes that must be made in the manipulated variables when load disturbances occur. Those pairings are recommended that result in changes that are not strongly affected by having other loops on automatic. Thus load disturbances are considered in the search for an eigenstructure. Tyreus (1987) has recently proposed a design procedure that combines the concepts of Buckley, Douglas, and Luyben. There are three steps: (1)Select several alternative material balance structures and evaluate the open-loop composition responses of each to various load disturbances. Select the material balance scheme that is the most effective in attenuating the disturbances. (2) Use steady-state optimization programs to determine which of the remaining manipulated variables should be set at some constraint for various load disturbances. This optimization can include minimizing operating costs, maximizing capacity, or maximizing profit. (3) Find simple policies for the remaining unconstrained manipulated variables, such as holding them constant or relating them in a feed-forward way to load disturbances. Thus the concept of eigenstructure fits well with this proposed procedure. Georgakis (1986) has proposed the control of calculated “extensive” variables such as reaction rate, total energy content, or total light component content instead of the traditional intensive variables (temperature, pressure, etc.). Manipulated variables can also be chosen to be sums, differences, or ratios of flow rates. The resulting eigenstructure handles disturbances more effectively.
Conclusions Eigenstructure is a useful concept in the analysis of multivariable process control systems. It can be applied to a variety of systems but should be particularly useful in the plant-wide control problem. Selecting control
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Tyreus, B. D. "Optimization and Multivariable Control of Distillation Columns". Advances In Instrumentation Vol. 42, Part 1, Proceedings of ISA 87 International Conference, 1987. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Haggblom, K. E. "On the Difference Between Distillation Column Control Structures". Report 86-2,1986; Process Control Laboratory, Abo Akademi, Abo, Finland.
structures on the basis of reducing loop interaction is not a valid criterion in most process control applications. The selection of the structure for a process control system should be made on its ability to effectively reject load disturbances.
Literature Cited Buckley, P. S. Techniques of Process Control; Wiley: New York, 1964. Fisher, W. R.; Doherty, M. F.; Douglas, J. M. "Synthesis of Steadystate Control Structures For Complete Chemical Plants, Part I11 Control Structure Synthesis Strategies". Paper 82b, presented at the AIChE Meeting, San Francisco, Nov 1984. Georgakis, C . Chem. Eng. Sci. 1986,41, 1471. Luyben, W. L. Ind. Eng. Chem. Fundam. 1975,14, 321. Niederlinski, A. AIChE J . 1971, 17, 1261. Stanley, G., Marino-Galarraga, M., McAvoy, T. J. Ind. Eng. Chem. Process Des. Dew. 1985, 24, 1181.
William L.Luyben Process Modeling and Control Center Department of Chemical Engineering Lehigh University Bethlehem, Pennsylvania 18015
Received for review May 14, 1987 Revised manuscript received October 14, 1987 Accepted November 2, 1987
Kinetics of the Pyrolysis of Chlorodifluoromethane T h e thermal decomposition of chlorodifluoromethane has been studied a t 1.379 X lo4-4.826 X lo4 P a (2-7 psig) over the temperature range 750-950 "C and residence times of 0.1-0.25 s, which is consistent with commercial practice. Within the limits of detectability, no perfluoroisobutylene was produced in these short residence time runs. Reaction mechanisms are proposed which include all products and side products identified in the reactor effluent streams by gas chromatography-mass spectroscopy. Arrhenius preexponential factors and activation energies are estimated for each of the reactions using a numerical integration scheme coupled to a Marquardt least-squares estimation algorithm. Operation of the tubular reactor was simulated a t three different temperatures using optimal estimates for the parameter values. Changes in product concentration along the tubular reactor then define operating conditions for any desired product mix. Aliphatic fluoroolefins are made commercially by the direct pyrolysis of monochlorodifluoromethane (FC-22) or the pyrolysis of other fluoroolefins. This study concentrates on the most important commercial fluoroolefins: tetrafluoroethylene (TFE) and hexafluoropropylene (HFP). Pyrolysis of FC-22 was studied in the past (Park et al., 1947; Gozzo and Patrick, 1966; Edwards and Small, 1965), but the temperature range in these studies (500-700 "C) and uncertainties in separation and identification of the pyrolysis products convinced us of the need to further investigate the kinetics of the pyrolysis. Our study covers the operating range 750-950 "C, which is consistent with commercial practice. This study eliminates the need to extrapolate kinetic coefficients reported in the literature beyond the range of their experimental data. Pyrolysis of FC-22 is carried out commercially in tubular reactors with residence times of 0.1-0.25 s. Thus study was carried out in a tubular reactor using the same range of residence times.
alumina tower and its composition analyzed. Analysis of the gas was done on a gas chromatograph (GC) with a Puracil B column. Components of the gas were identified by using GC mass spectroscopy; thus trace amounts of fluorocarbons were identified and quantified.
Experimental Section Figure 1 is a schematic description of the experimental unit. CHClF2 (GENETRON-22, Allied Corporation) was supplied from a cylinder through a rotameter a t a constant pressure to a 0.95-cm (3/s-in.) i.d. tubular reactor made of Inconel. The reactor was heated with three Lindberg single-zone tube furnaces. The first furnace of 30 cm (12 in.) was used as a preheater and the other two, with a total length of 60 cm (24 in.), defined the pyrolysis section. The gas exiting from the reactor was immediately quenched in a helical double-wall water-cooled heat exchanger. The gas was cooled below 400 "C, thus stopping any further reaction. After cooling, the pyrolysate was scrubbed with water or HCl solution to remove the HCl and dried in an
CzF4+ H C 1 5 H(CF2),Cl
Reaction Mechanisms The kinetics of the pyrolysis can be described by the following set of equations: CHClF2 7 CF2' 1
+ HCl
3 7 C2F4 2C2F4 7 5 c-C~F~
2CF2'
C2F4+ CF2' 9
I
(A)
(B) (C)
C3F6
(E)
These relations describe the major reactions taking place. A series of high boilers with a general molecular formula H(CF,),Cl are produced, but their concentration per pass is low. Normal commercial practice is to recycle the high boilers. In this study we assumed that they were part of the H(CF2),C1 (FC-124A) concentration. The key reaction of the pyrolysis is the generation of CF2' radicals, which by recombination create the desired fluoroolefins. The kinetics of each reaction was assumed to conform with an Arrhenius temperature dependence. The pyrolysis was carried out at isothermal conditions and the temperature along the reactor was monitored and 0
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