Znd. Eng. Chem. Res. 1987,26, 2099-2106
Use of Multiplicity Features for Kinetic Modeling: Oxidation on Pt/Al,O,
2099
co
Michael P. Harold* and Dan Luss University of Houston, Department of Chemical Engineering, Houston, Texas 77004 Experimental, nonisothermal, steady-state multiplicity features of a self-inhibiting catalytic reaction, CO oxidation on a single Pt/A1203 pellet, are analyzed and modeled. Inspection of the bifurcation diagrams and map indicates that thermokinetic coupling and interparticle transport limitations affect this system. A lumped-thermal steady-state model is used to fit the location of the observed ignition and extinction points. All the observed trends are adequately predicted, but the quantitative fit is inadequate at low pellet temperatures. This is probably due to the use of an oversimplified kinetic model or an inadequate estimate of the activity profile. T h e model properly predicts a transition from a thermokinetic to an isothermal multiplicity mechanism. The simulations show that a state which is stable under isothermal conditions may be destabilized by thermal factors. 1. Introduction
Steady-state multiplicity is exhibited by many catalytic reactions. This behavior may be caused by different mechanisms including a thermokinetic interaction (Aris, 1975 and references within), isothermal coupling between a self-inhibiting reaction and either intraparticle diffusion limitations (Elnashaie and Yates, 1973; Becker and Wei, 1977a,b) or interparticle transport limitations (Matsuura and Kato, 1967; Beusch et al., 1972),or the coupling among two or more steps in the overall reaction sequence (Slinko et al., 1972; Pikios and Luss, 1977; Eigenberger, 1978; Slinko and Slinko, 1978; Takoudis et al., 1981a,b; Bykov and Yablonski, 1981a,b; Kevrekidis et al., 1984). The qualitative multiplicity features may be used to predict the simplest functional form a rate expression needs to have in order to fit the data (Harold et al., 1987a,b). These features may be displayed in bifurcation diagrams which describe the dependence of the system on a slowly changing operation condition (bifurcation uariable) and/or in a bifurcation map which describes a plane of two operating conditions regions with different numbers of solutions. Moreover, the ignition and extinction (bifurcation) points may be used to estimate the kinetic parameters (Harold and Luss, 198713). We examine here the possible use of multiplicity information to predict the intrinsic rate expression for a self-inhibiting, exothermic catalytic reaction for which the diffusional limitations cannot be neglected. The ignition and extinction points were found experimentally during the oxidation of carbon monoxide on a single Pt/A1203 catalytic pellet. The analysis of such data is essential for assessing the applicability and potential pitfalls of the use of experimental ignition and extinction points to fit kinetic data. We describe first the experimental observations. We then find the simplest rate expression which can predict the observed qualitative features. Finally, we estimate the kinetic and transport parameters which fit best the bifurcation points and discuss the implications of predicted bifurcation diagrams. 2. Experimental Observations of Steady-State
Multiplicity Carbon monoxide was oxidized on a single 0.3% w t Pt (wt and v refer to by weight and by volume, respectively) m diameter, made on alumina spherical pellet of 4 X by Oxy-Catalyst (division of Met-Pro Corporation). Most
* Present address: Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003. 0888-5885/87/2626-2099$01.50/0
of the Pt in the fresh pellets was concentrated in a thin m) exterior spherical shell (see Figure 1in Harold and Luss (1987a)). Figure 1shows the Pt profile, measured by a scanning electron X-ray microprobe, of a pellet used for several months. Clearly, considerable redistribution of the Pt occurred, and the precious metal content decrease almost linearly from its surface value within an exterior shell of 5.0 X m thickness. The Pt is distributed uniformly in the inner core with a concentration of about 20% of that at the surface. A mixture of oxygen, nitrogen, and carbon monoxide flowed past the pellet with a linear velocity of about 1.55 X m/s. The center pellet temperature, Tp, was and the measured as a function of the gas temperature, Tg, CO mole fraction, xb. A detailed description of the experimental setup and procedure was presented by Harold and Luss (1985). Multiple steady states were observed for all CO concentrations in the range of 0.64% v and some gas temperatures in the range of 20-250 "C. The bifurcation diagrams of Tp vs. TFfor a fixed xb, typical examples of which are shown in Figure 2, exhibited a counterclockwise hysteresis. At most, two stable states were found for any set of operating conditions. For X b 5.44% v Co. This is probably due to a thermal destabilization of the state with the high rate, i.e., the one with the lowest Tgfor a specified Tp. These states should be observable (stable) when the pellet and gas temperatures are the same, Le., isothermal operation. Figure 6 compares the observed and predicted (TP,x b ) region in which inverse multiplicity was found. The predicted region has the shape of a positively sloped cusp. The model predicts that inverse multiplicity exists for any pellet temperature exceeding 130 "C (cusp point). However, in the experiments, inverse multiplicity was not observed for any pellet temperature larger than 215 "C. This suggests a destabilization of states which would be stable during isothermal operation. Thus, the region of an inverse multiplicity of two stable states is closed. We show in Figure 6 also the region of isothermal rate multiplicity observed by Beusch et al. (1972) for the oxidation of carbon monoxide on a single Pt/A120, pellet. Interestingly, their data reveal the same qualitative fea-
200
180
0
1
0
160 160
220
('CC)
Figure 7. Dependence of the effectiveness factor along a simulated bifurcation diagram.
tures as predicted by our model. In their experiments, the linear velocity was sufficiently high to keep the pellet temperature essentially equal to that of the gas. This reduction in the thermal resistance probably prevented the thermal destabilization a t high reactant concentrations, which our model predicts. It is of interest to assess the impact of the intraparticle diffusional limitations on this pellet in which most of the catalytic activity is within a narrow exterior shell. Figure 7 shows the computed effectiveness factor for various points along the simulated Tpvs. Tgbifurcation diagram for xb = 5.75% v. The effectiveness factor is slightly larger than unity along the low-temperature branch. Under these conditions, the intrinsic rate is relatively low due to the strong reactant inhibition, and the diffusional reduction in the reactant concentration increases the rate a t the center of the pellet over that at the surface. Along the ignited branch, the effectiveness factor is a monotonic decreasing function of pellet temperature and is smaller than unity for Tplarger than about 250 OC. The effectiveness factor decreases drastically for Tp > 300 "C, indicating a strong impact of diffusion limitations even though the catalytic activity is contained mostly in a thin surface shell. The high effectiveness factor noted for T p< 220 OC is due to the coupling of the self-inhibited kinetics with intraparticle diffusion. We note that for Tp < 220 "C, the inverse multiplicity implies the existence of two stable rates for a constant pellet temperature. Again, this is caused by the isothermal coupling between the kinetics and interparticle and/or intraparticle mass-transfer limitations. 6. Concluding Remarks
This study demonstrates the utility of examining in detail both qualitatively and quantitatively the steadystate multiplicity features of a catalytic reaction. The method is useful for determining the functional features of the intrinsic kinetic model and for estimating the model kinetic and transport parameters. For reactions which exhibit steady-state multiplicity, this method has several distinct advantages over traditional kinetic studies. First, the finding of the multiplicity data involves a straightforward experimental design and operation with fewer experimental constraints. The single pellet reactor used to obtain the CO oxidation data enabled a rapid acquisition of data over a wide range of operating conditions and a sensitive determination of the bifurcation point locations (see Harold (1985) for a detailed description). A temperature rise between the catalyst and the bulk re-
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2105 acting mixture due to the reaction exothermicity is a desired feature with our approach, as the heat effects may lead to thermokinetic multiplicity. On the other hand, temperature gradients in a differential or integral reactor are usually undesirable, and therefore attempts are made to minimize them. However, a complete elimination of thermal gradients is difficult for highly exothermic reactions. Our method exploits the ignition and extinction phenomena for kinetic model discrimination. Second, our method involves a single systematic set of experiments which enables both the intrinsic kinetic model and parameters to be determined. The qualitative tests of the single pellet data rapidly eliminate a large class of kinetic models based on their inability to predict the bifurcation diagrams and map features. By exploiting multiplicity as a means of kinetic analysis, this method provides crucial information about catalyst performance under conditions of extreme sensitivity to the operating conditions. Traditional kinetics analysis methods do not probe the conditions which give rise to reaction-transport-induced multiplicity simply because the heat and mass transport resistances are minimized. Hence, another set of experiments must be conducted which investigate the impact of the multiplicity on catalyst performance. Third, discrimination among kinetic models which pass the qualitative tests involves a check of their fit of the ignition and extinction data. The ability of a mathematical model to fit the bifurcation data helps to elucidate the complex contribution of several multiplicity inducing mechanisms on the overall catalyst performance. The best fit steady-state model can then be incorporated into a reactor model. An adequate fit of the CO oxidation multiplicity features was achieved with a model incorporating a bimolecular Langmuir-Hinshelwood rate expression, interparticle and intraparticle transport resistances, and an estimate for the intraparticle activity profile. While the kinetic model does not account for all the complexity of the CO oxidation kinetics, all of the qualitative trends are successfully predicted. The conservative prediction of the multiplicity region for low reactant concentrations and pellet temperatures suggests that an intrinsic kinetic mechanism may have contributed to the multiplicity. Lack of accurate knowledge about the instantaneous activity profile prevented the testing of intrinsically multivalued models, which contain a larger number of parameters. Moreover, little is known about catalyst performance when a reaction which can exhibit intrinsic rate multiplicity is coupled with transport resistances. Future studies should focus on this issue as well as the occurrence of spatial nonuniformities observed in other works (e.g., Cox et al., 1985; D’Netto et al., 1984; Kaul and Wolf, 1984, 1985a,b). We note that it may be difficult to model in an exact quantitative sense the steady-state features of a catalytic pellet with a nonuniform intraparticle activity distribution. As we have shown in another study (Harold and Luss, 1987a), the active component in a surface shell-type catalytic pellet can migrate away from the surface. The redistribution can dramatically alter the functional dependence of the overall rate on the limiting reactant concentration for a reaction with self-inhibitive kinetics. Thus, we suggest to use uniformly active catalysts in order to avoid these complicating effects. A more complete understanding of this active component migration is needed. Acknowledgment We thank the National Science Foundation, the Welch Foundation, and Shell Oil for financial support of this
research. We also acknowledge General Motors Research Laboratories and Shell Development for providing the electron microprobe scan of the catalyst pellets. Nomenclature a = dimensionless activity A = activity parameter defined in (28) C = intraparticle concentration De = effective diffusivity E = activation energy E, = expected value F = function defining steady state h = heat-transfer coefficient H = Heaviside function AH = heat of reaction AHa = heat of adsorption k = rate constant k = volume average rate constant k , = mass-transfer coefficient K = adsorption equilibrium constant L = dimensionless adsorption constant defined by (18) p = parameter pT = total pressure q = response Q = dimensionless adsorption constant defined by (11) r = reaction rate r’ = intraparticle radial distance R, = ideal gas constant R, = pellet radius R1 = radius of inner inactive core s = dimensionless intraparticle radial distance s1 = dimensionless radius of inactive pellet core s2 = dimensionless width of linearly increasing activity region S = sum of squares S,= external surface area Sh = Sherwood number T = temperature T = average film temperature AT,, = adiabatic temperature rise V , = pellet volume w = weight factor x = mole fraction y = dimensionless pellet temperature z = dimensionless surface concentration
Greek Symbols
dimensionless gas temperature p = dimensionless adiabatic temperature rise given by (18) p’ = modified dimensionless adiabatic temperature rise given by (14) e = error between measured and predicted temperatures y = dimensionless activation energy Y~ = dimensionless heat of adsorption 9 = isothermal effectiveness factor 4 = Thiele modulus 9 = Weisz-Prater modulus a=
Subscripts
a = adsorption ad = adiabatic b = bulk c = mass transfer g = gas p = pellet s = surface 0 = reference Registry No. CO, 630-08-0; Pt, 7440-06-4.
Literature Cited Aris, R. The Mathematical Theory of Diffusion and Reaction i n Permeable Catalysis; Clarendon: Oxford, 1975. Becker, E. R.; Wei, J. J . Catal. 1977a, 46, 35. Becker, E. R.; Wei, J. J. Catal. 1977b,46, 372.
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Beusch, P.; Fieguth, P.; Wicke, E. Chem. Ing. Technol. 1972,44,445. Bykov, V. I.; Yablonski, G. S. Int. Chem. Eng. 1981a,21, 142. Bykov, V. I.; Yablonski, G. S. Surf. Sci. 1981b,107, 1334. Cant, N. W.; Hicks, P. C.; Lennon, B. S. J . Catal. 1978,54, 372. Cochran, H. D.; Donnelly, R. G.; Modell, M.; Baddour, R. G. Colloid Interface Sci. Proc. 50th Znt. Conf. 1976,3, 131. Cox, M. P.; Ertl, G.; Imbihl, R. Phys. Rev. Lett. 1985,54, 1725. D'Netto, G. A.; Brown, J. R.; Schmitz, R. A. Proc. 8th ZSCRE, Pergamon: New York, 1984; Vol. 248. Eigenberger, G. Chem. Eng. Sci. 1978,33,1255. Elnashaie, S.S.;Yates, J. G. Chem. Eng. Sci. 1973,28,515. Engel, T.;Ertl, G. Adv. Catal. 1979,28, 1. Froment, G. F.; Hosten, L. "Catalytic Kinetic Modelling", In Catalysis Science and Technology;Anderson, R. R., Boudart, M., Eds.; Springer-Verlag: New York, 1984; Vol. 2, p 97. Furusawa, T.; Kunii, D. J. Chem. Eng. Jpn. 1971,4 , 274. Harold, M. P. Ph.D. Dissertation, University of Houston, 1985. Harold, M. P.; Luss, D. Chem. Eng. Sci. 1985,40,39. Harold, M. P.; Luss, D. Znd. Eng. Chem. Res. 1987a,26, 1616. Harold, M. P.; Luss, D. Znd. Eng. Chem. Res. 198713,in press. Harold, M. P.; Sheintuch, M.; Luss, D. Znd. Eng. Chem. Res. 1987a, 26,786. Harold, M. P.; Sheintuch, M.; Luss, D. Ind. Eng. Chem. Res. 1987b, 26, 794. Heck, R. H.;Wei, J.; Katzer, J. R. AZChE J . 1976,22,477. Hegedus, L.L.; Oh, S.H.; Baron, K. AZChE J. 1977,32, 632.
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Received for review December 5, 1986 Accepted June 15, 1987
Control of Constrained Multivariable Nonlinear Process Using a Two-Phase Approach Shi-Shang Jang,+Babu Joseph,*+and Hiro Mukail Chemical Engineering Department and Department of Systems Science and Mathematics, Washington University, S t . Louis, Missouri 63130
A computer-based algorithm is presented for the control of complex process units which are characterized by difficult features such as nonlinear input/output relationships, multivariable nature, operational constraints, imprecise models, and inadequate measurements. A two-phase algorithm is proposed to deal with these difficult features in a direct manner. The algorithm makes use of approximate process models which take into account significant physical and chemical events in the process. The first phase of this algorithm consists of identifying unmeasured disturbances and the parameters and states of the approximate model using measurements from the immediate past. This identified model is then used in the second phase to predict the future behavior of the process and to select an appropriate control action. Application of the algorithm to a highly nonlinear, difficult-to-control CSTR system shows the advantages of this approach over other approaches which are based on linear models. Robustness in the presence of plant/model mismatch and immunity to measurement noises are also illustrated. Most regulatory control problems in the process industry are handled well by the traditional PID-type feedback controllers which are relatively inexpensive and well understood. However, there are a number of complex process operations which still present challenging problems in control for a variety of reasons. The reasons most often cited in the literature (Foss, 1973) include the following. (i) Process Nonlinearity. This limits the applicability range of controllers based on linear models. (ii) Multivariable Nature. A unit may have more than one control variable and more than one output variable to be controlled. (iii) Inadequate Measurements. The variables to be controlled may not be directly measurable on line. In such a case, the process must be controlled by using secondary measurements. (iv) Process Constraints. The control system must maintain process variables within certain bounds using Chemical Engineering Department.
* Department of Systems Science and Mathematics.
bounded inputs. Moreover, it must be able to deal with these constraints even when the constraints change with time. Modern control theory, developed in the 1960s, has not found much acceptance by the process industry, primarily because the theory did not address many of the above issues which were critical to the process industry. However, the availability of on-line process computers spurred the development of many new algorithms for computer control. Examples are Dynamic Matrix Control developed at Shell (Culter and Ramaker, 1979), Model Algorithmic Control developed in France (Richalet et al., 1978), and Inferential Control (Brosilow, 1979). Garcia and Morari (1982) discussed these and similar algorithms from a theoretical point of view and coined a new name, Internal Model Control, to represent this class of algorithms. The success of these algorithms can be attributed to the following important features: (i) relative ease in generating the process model, (ii) the ability to utilize process measurements to compensate for unmeasured disturbances and modeling errors, and (iii) the use of a filter in the feedback
0888-5885/87/2626-2106$01.50/0 0 1987 American Chemical Society
