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Kinetics Falsification by Symmetry Breaking. 2. Olefin Oxidation on a Platinum Wire Moshe Sheintuch,* Judith Schmidt, and Shlomo Rosenberg Department o f Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
T h e multiplicity patterns of olefin oxidation, catalyzed by a Pt wire controlled to maintain a preset average temperature, are traced by varying the reactant concentrations and are mapped in the concentration plane. T h e similar bifurcation maps of ethylene and propylene oxidations exhibit regions of tristability and isolated branches in the directions of olefin and oxygen concentrations. Isobutylene oxidation exhibits mushroom-shaped bistability. Several features indicate the occurrence of symmetry breaking t o form a partially ignited wire. Comparison of these results with the characteristic bifurcation maps of inhomogeneous solutions, drawn in part 1 of this work, shows that these complex patterns are induced by coupling of reactant inhibition with strong exothermicity. This work also develops an efficient methodology for fast tracing of all the multiplicity features of reacting systems. Bifurcation diagrams are traced automatically with a microcomputer-governed experimental system, and the continuity of features in the second and third dimensions is studied off-line t o decide on further experimentation. In the first part of this work (Sheintuch, 1989), we demonstrated that surprisingly complex multiplicity patterns can be induced by relatively simple kinetics in systems that attain stable inhomogeneous solutions. We reviewed previous observations of bistability in reactions catalyzed by an “isothermal” wire, i.e., controlled to maintain a preset (average) temperature, and showed that the patterns with counterclockwise hysteresis or with isolated branches can be accounted for by stationary thermal fronts in systems with simple kinetics or with reactant inhibition. The analysis was also extended to systems that show tristability in order to account for the observations reported in this part. This work presents a comparative study of multiplicity patterns and bifurcation maps in olefin oxidation on a Pt wire, using a thermochemic method. Regions of uniqueness, bistability, and tristability are mapped in the oxygen versus olefin concentration plane at several temperatures. The analysis shows that even these complex patterns can be explained by simple kinetics due to the control that stabilizes stationary fronts. We emphasize the invalidity of the methodology for model discrimination in bistable systems (Harold et al., 1987), which assumes uniformity of all solutions and is based on the bifuraction properties of algebraic equations. The importance of this conclusion extends beyond direct application to the thermochemic method: Symmetry breaking may occur along a packed bed catalyzing an exothermic reaction (Sheintuch, 1987) or a cylindrical pellet catalyzing an endothermic reaction (Hefer and Sheintuch, 1986). Since the experimental detection of such states is still a complex task, while an analytical proof of their existence is subject to complex stability analysis, it is important to identify qualitative integral features that are unique to inhomogeneous solutions. We emphasize these features in our analysis. The multiplicity patterns (bifurcation diagrams) are traced automatically in this work, using microcomputergoverned experimental systems, by sweeping the concentration of one reactant back and forth. To trace all possible diagrams, the process is repeated at various concentrations of the second reactant. The locus of ignition and extinction points is mapped then in the reactant concentration plane. The methodology of sampling is detailed in the next section. In the Results section, the bifurcation diagrams and maps are outlined in the domain 0-270 olefin and 0-50% 0888-5885/89/2628-0955$01.50/0
O2 to elucidate the singular points and to compare the multiplicity features of the Cz-C4 olefin oxidations. Bistability was reported for the oxidations of ethylene (Zhukov and Barelko, 1976) and propylene (Sheintuch and Luss, 1983) on an isothermal Pt wire. Both studies traced the bifurcation set in the temperature-olefin concentration plane, but differences in oxygen concentration and activation procedures do not allow a direct comparison between the two. Our study shows up to three solutions organized in patterns significantly more complex than those reported earlier; furthermore, we show that the bifurcation maps of these two reactions are similar. The analysis in the Discussion section shows that the bifuraction map can be accounted for by inhomogeneous solutions of a wire that sustains an exothermic reaction characterized by kinetics with reactant inhibition (e.g., a Langmuir-Hinshelwood rate expression). We also identify the experimental features that indicate the possible existence of inhomogeneities. For comparison, we also outline the kinetic features of a lumped model that accounts for the data showing the unnecessary complexity.
Methodology and Experimental Section The optimal design of experiments for model discrimination purposes is based on sequential sampling at points in the parameter space that will yield the largest divergence between the predictions of the proposed models. These predictions are made with parameter values that are reestimated after each sampling. Strategies for the optimal conduction of experiments. have been suggested by Box and Hill (1967) and by Froment and co-workers (Froment and Bischoff, 1979; Froment and Hosten, 1984). The latter have applied it off-line to multireaction networks. On-line real-time applications have been demonstrated by Scotting et al. (1974) and by Mandler et al. (1983). A unique steady state was expected and observed in all the studies above. Applications of these strategies for multivalued rate curves would require (a) an efficient methodology for scanning all the branches, (b) identification of the various branches in the experiments as well as in the model predictions, and (c) a fast and reliable parameter estimation procedure. The methodology of the parameter estimation of multivalued functions is currently emerging (Harold and Luss, 1987; Sheintuch and Avichai, 1988) and is still too complex for application after every sampling point. 0 1989 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 W
GLL
Figure 1. Several bifurcation diagrams (upper row) and sets (lower row) that cannot be traced by varying CI.
We develop now an efficient methodology to trace all the existing rate branches within the operation domain and to move from one branch to another in a systematic way. When the solution is unique everywhere, then the order of sampling points is solely determined by the optimization condition for the model discrimination or parameter estimation process. When only two stable branches exist and are organized in a simple (counterclockwise or clockwise) hysteresis pattern, then sweeping one of the concentrations (say C1, Figure la) back and forth will reveal the ignition and extinction points, if both are within the operation domain. When the process is repeated for other fixed values of C,, the bifurcation set will be traced revealing a cusp-shaped region (Figure 1b). The identity of the primary (C,) and secondary (C,) variables and the direction of the sweeping cycle of C, (i.e., high to low to high concentrations or the opposite way) are not important except in the range with ignition or extinction points outside the operating region. Sweeping C1 a t fixed C2 > C2* will reveal only the upper branch since the spontaneous extinction point is not achieved (Figure lb). Exchanging the primary and secondary directions does not remedy the situation (for the set shown in Figure lb) since the extinction points with C , > C1* escape detection then. T o find both branches, it may be necessary to conduct experiments in both directions or to force extinction a t known conditions, a t the end of every sweeping cycle, before decreasing C , from high to low concentrations. When the ignition and extinction lines are of opposite slopes (Figure Id), then sweeping C, reveals very little information. For C2 > C2*, the upper branch is continuous (e.g., Figure IC)and the extinction point will not be detected. Similarly, for C z < C2* the lower branch is continuous. The problem may be remedied by exchanging the primary and secondary directions. The bifurcation diagram with varying C2 is S shaped ( C , < C,*). Similar problems and remedies exist when a closed isolated branch (isola) exists in one direction. The problem is even more complex when three or more stable solutions exist, since sweeping one concentration back and forth cannot reveal all three branches. No systematic approach can be suggested now since dynamics dictate the fate of transitions under conditions where two other stable branches exist. Furthermore, branches that appear as a nested loop within the larger hysteresis curves (Figure l e i will escape detection. The corresponding section of the bifurcation set (broken line, Figure If) will not be revealed in either direction. The strategy adopted in this experiment assumes the following steps: (a) Conduct automated experimentation by sweeping one concentration (usually the fuel) as the primary variable and the other as the secondary. (b) Draw the bifurcation set, and check for its continuity and consistency of the diagrams. (c) If the set is incomplete, repeat the first step with exchanged roles of concentrations. (d) If the resolution of the bifurcation set is insufficient, then
the previous step(s) may be repeated over a smaller domain with smaller steps. (e) The information obtained a t this stage usually was found to be sufficient except a t certain parameter values with a nested loop or discontinuous branches that were not attained in either experiment. The desired branch was achieved, then, manually, and the automated experimentation continued in one dimension only. To initiate the automated search, the user needs to supply the following technical information: (a) boundaries of the experimentation domain; (b) mode of operation (the primary and secondary variables and their initial directions); (c) sampling intervals; (d) conversion factors and technical constants. From then on, the algorithm is completely independent. Steady state is assumed to be attained when good reproducibility is found over a predetermined number of readings. When sweeping back a variable, readings are compared with those taken in the main direction, and steady state is assumed when their difference is small ( Y
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Figure 8. Bifurcation diagrams with varying oxygen concentrations (sampling intervals of 5%);the diagrams with 1% CzH4or more were traced after forced ignition of the system.
was constructed mainly from data obtained by employing ethylene concentration as the primary variable. In that direction, the upper and middle branches are traced since they are organized in a simple hysteresis loop (Figure 6, 27-47% 02):Transition from the upper branch leads to the intermediate one and vice versa. The inactive branch is isolated in this range. Once it is established, either by forced extinction or by changing the mode of operation, then decreasing fuel concentration leads to ignition to the upper branch. The nil activity branch is attained spontaneously only below 22% 02. The inactive and active branches are organized in a simple hysteresis loop. Between these two characteristic diagrams and at sufficiently large C1,there exists an isolated ignited branch. Decreasing ethylene concentration along this branch induces extinction to the nil activity branch a t 17-22%. The mechanism by which the isolated branch is fused into the clockwise hysteresis loop is unclear. One possible mechanism, which applies to a lumped system, is shown by the assumed structure of the unstable branches (denoted by a broken line, Figure 6). Employing oxygen concentration as the primary variable reveals very little information. Spontaneous ignition oc-
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Figure 10. Bifurcation set for propylene oxidation a t two temperatures.
curs, within the operating domain, only below 0.7% C2H4 (Figure 8). Forcing ignition a t the beginning of a run is not rewarding either since it reveals only one active branch; the other branch is either isolated (with more than 1.2% C2H4)or forms a nested loop. The bifurcation diagrams and amps of propylene oxidation (Figures 9 and 10) are similar to those of ethylene, showing a transition between two clockwise hysteresis loops: At high O2 concentration, the middle and upper branches are organized in such a loop (26% and 22% 0 2 (not shown) at 240 "C and 42% a t 219 "C), while at low oxygen concentrations the loop combines the active and inactive branches (7% and 17% O2at 240 "C, 17% O2(not shown) at 219 "C). The transition between these diagrams occurs through an isolated branch (27% at 219 "C) that disappears with declining oxygen concentration. A t 219
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 959 and it exists even a t 2% C3H6 (Figure 9, 32%). The bifurcation set (Figure 10) was constructed again by sweeping fuel concentration. The diagrams obtained in the oxygen direction reveal only one active state. The other branch of stable active states is not accessible.
Discussion In comparing the three reactions in this study, we focus our attention on the features to be elucidated from the bifurcation diagrams and maps. The bifurcation map of isobutylene oxidation differs from those of ethylene and propylene oxidations: multiplicity in the former is limited to bistability, compared with tristability in the latter, and no apparent isolated branches were observed in the former. The feature common to the three reactions is the hysteresis loop in the clockwise direction between the active and inactive branches. Such a bistability pattern is usually attributed to reactant inhibition, a feature that is common to many catalytic reactions. Comparison of these regions in the three reactions at 240 "C shows that the bistability domains in the CzH4and C3H6cases lie in the region of excess oxygen, while in C4H8 oxidation the multiplicity appears in slightly fuel-rich mixtures. The latter cannot be accounted for by the bimolecular explicit LangmuirHinshelwood rate expression and may be of a different origin. We show later that indeed the isobutylene loop is induced by symmetry breaking. Before we turn to the analysis of inhomogeneities, let us consider the features that suggest the existence of such solutions. The maps of C4H8oxidation (Figure 4) can be completed to show two independent cusp points at 173 and 193 "C. Continuity of features in the third parametric dimension, as the temperature is varied to 240 "C, suggests that the bistability domain is continuous. The map is then similar to the map drawn for inhomogeneous solutions in reactions with single-valued self-inhibited kinetics, like the Langmuir-Hinshelwood rate expression. The bifurcation map corresponding to this situation, in the plane of the average temperature (T,) versus the inhibiting reactant concentration (C,),is shown in Figure I d of part 1. Transforming the map to the concentration plane (C, versus C,, Figure 4) should not change its nature or the orientation of boundaries, since the temperature and oxygen concentration have similar effects: their increase accelerates the reaction rate. The positive dependence on oxygen is evident at 173 or 193 "C (Figure 31, while at 240 "C it had an inhibitory effect only outside the bistability domain. We conclude, therefore, that the bistability domain in Figure 4 is continuous and bounded from above by ignition points from the homogeneous inactive branch, while the lower bound is the loci of extinction points from the branch of partially ignited states. That interpretation is also supported by the proximity of the two rate branches a t extinction, especially at low concentrations (Figures 2 and 3): in an infinitely long wire and in the absence of cooling effects at the edges, the two branches should touch. That interpretation suggests that the cusp point (Figure la, part 1)lies outside the operation domain a t 173 and 193 "C. We turn now to the complex maps of ethylene and propylene oxidations. Two cusps are clearly identified at the points where each of the two counterclockwise hysteresis loops disappears (denoted by a and e in Figures 7 and 10). The layout of these cusps is similar to the characteristic bifurcation set of an algebraic equation admitting tristability, but another cusp point is missing. It should be probably added around the merging point of the isolated branch (unshaded area in Figures 7 and 10). This
will be in agreement with the disposition df the unstable branches in Figure 6 but in contradiction with the shape suggested by the bifurcation set showing a smooth line. A clue to the existence of inhomogeneous solutions is provided by the proximity of the upper and the intermediate branches (Figures 6 and 9). We also find a local extremum in the oxygen direction for both reactions a t 240 "C (denoted by d), close to the self-intersection of the bifurcation set (point c). We have shown in part 1 that this feature is characteristic to inhomogeneous solutions. The similarity of the bifurcation maps of CzH4 and C3H6 oxidations to the map in Figure 6, part 1, is evident, and points a, c, and d are marked in both maps. Mapping from the temperature ordinate in the latter to the oxygen ordinate to the former should not change the shape of the bifurcation set (but may change its orientation) since the rate ascents with oxygen (Figure 8). The similarity suggests that the observations are due to symmetry breaking in a system with strong reactant inhibition, which induces isothermal multiplicity.
Concluding Remarks This work demonstrates again that the thermochemic method is an unsuitable mode for kinetic study of highly exothermic reactions. The control does not render the system isothermal but rather stabilizes the asymmetry in the system. As a result, the controlled system acquires many of the features of the nonisothermal-uncontrolled system. An (uncontrolled) exothermic self-inhibited reaction may exhibit tristability and discontinuous branches of steady states. Similar features are observed here (Figures 6-10), and accounting for them by an isothermal model requires an unnecessarily complex mechanism. The false kinetics implied by observations in a controlled system extends to studies into reaction dynamics as well: a nonisothermal oscillator will be translated under control into apparently isothermal oscillations in which several sections of the wire oscillate out of phase-and maintain the required set point. Analysis of this situation shows that the interaction of these sections may lead to multipeak oscillations. Thus, aperiodic behavior may be induced by such a control. The shortcoming of applying control to distributed parameter systems extends beyond the thermochemic method. Antiphase oscillations were recently reported for electrochemical reactions under galvanostatic (constant current) control (Lev et al., 1988). In swdies of laboratory adiabatic reactors, it is customary to apply a certain amount of heating to the reactor walls in order to compensate for the heat loss, which is almost always unavailable. If a front is formed within the reactor, then improper control of such heaters may stabilize the front position. These arguments suggest that symmetry breaking may be more commonly prevalent than previously thought. This work also develops an efficient methodology for automated tracing of multivalue rate curves. The advantage gained by automated experimentation, aside from saving time, is the ability to plot the whole bifurcation set with minimal changes in catalyst activity since the wire is continuously exposed to reaction conditions. That is an important feature if the data for constructing the set are obtained at various modes, Le., with varying fuel or varying oxygen concentrations. We feel that the present combination of on-line and off-line decisions is almost optimal. Improved accuracy of transition points can be attained by a high-resolution experiment after the transition points have been identified in the low-resolution experiment, as
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we have shown in a recent work (Sheintuch and Avichai, 1988). The main obstacle for obtaining an efficient mapping strategy is the existence of isolated branches and nested hysteresis loops. To overcome the former problem, the experiment should be conducted in another parametric dimension. Designing an algorithm that will trace a nested loop is more difficult, as it requires changing two parameters simultaneously.
Acknowledgment This work was supported by the Israel Academy of Science and Humanities. Registry No. C2H4, 74-85-1; C3H6, 115-07-1; C4H8,115-11-7; Pt. 7440-06-4.
Literature Cited Box, G. E. P.; Hill, W. J. Discrimination Among Mechanistic Models. Technometrics 1967, 9, 57. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979; Chapter 2. Froment, G. F.; Hosten, L. Catalysis Science Technology; Anderson, J. R., Boudart, M., Eds.; Wiley: New York, 1984; Vol. 2, p 97. Harold, M. P.; Luss, D. Use of Bifurcation Map for Kinetic Parameter Estimation. 1. Ethane Oxidation. Ind. Eng. Chem. Res. 1987, 26, 2092. Harold, M. P.; Sheintuch, M.; Luss, D. Analysis and Modelling of Multiplicity Features. 2. Isothermal Experiments. Ind. Eng. Chem. Res. 1987, 26, 794.
Hefer, D.; Sheintuch, M. Multiplicity Patterns of Inhomogeneous Solutions in Catalysts with Self-Inhibited Reactions. Chem. Eng. Sci. 1986, 41, 2285-98. Lev, 0.; Sheintuch, M.; Pismen, L. M.; Yarnitski, H. Standing and Propagating Wave Oscillations in the Anodic Dissolution of Nickel. Nature 1988, 338, 458. Mandler, J.; Lavie, R.; Sheintuch, M. An Automated Catalytic System for the Sequential Optimal Discrimination between Rival Models. Chem. Eng. Sei. 1983, 38, 979-990. Schmidt, J.; Sheintuch, M. The Characterization of Kinetics by Singular Points: Comparison of Supported vs. Unsupported and Isothermal vs. Nonisothermal Pt Catalysts in Ammonia Oxidation. Chem. Eng. Commun. 1986,46, 289-309. Scotting, D. S.; Cowsley, C. W.; Mitchell, F. R. G.; Kenney, C. N. Computer Controlled Experimentation. Trans. Inst. Chem. Eng. 1974, 52, 349-353. Sheintuch, M. The Determination of Global Solution from Local Ones in Catalytic Systems Showing Steady State Multiplicity. Chem. Eng. Sei. 1987,42, 2103-2114. Sheintuch, M. Kinetics Falsification by Symmetry Breaking. 1. Steady-State Analysis. Ind. Eng. Chem. Res. 1989. preceding paper in this issue. Sheintuch, M.; Avichai, M. Design of Experiment and Parameters Estimation in a Bistable System: Ethylene Oxidation on Pt. Ind. Eng. Chem. Res. 1988,27, 1152-1157. Sheintuch, M.; Luss, D. Application of Bifurcation Theory to Modeling of Steady State Multiplicity: Propylene Oxidation on Platinum. Ind. Eng. Chem. Fundam. 1983,22, 209-215. Zhukov, S. A.; Barelko, V. V. Nonuniqueness of Stationary States in a Catalyzer and Flickering in the Oxidation of Ethylene on Platinum. Dokl. Akad. Nauk. SSSR 1976, 229, 655. Received for revieu May 2, 1988 Revised manuscript received January 24, 1989 Accepted February 22. 1989
Comparison of Various Models for a Gas-Organic Solid Reaction: Application to the Chlorination of Phenols Denis Pattou and Robert Perrin* Laboratoire de Chimie Industrielle, CNRS U A 805, UniversitP Claude Bernard, Lyon I , 43 Boulevard d u 11 Novembre 1918, 69622 Villeurbanne, France
Various models have been proposed in the literature in order to interpret solid-state reactions. These models are presented here and are applied to the study of the chlorination of solid 4-chlorophenol. T h e kinetic law seems to be, for powders, a first-order one (Cardew's theory); for single crystals, the model developed by Prout and Tompkins is the best adapted to experimental results. These observations lead to the conclusion that the organic-solid-state reaction occurs initially in some specific points that could be defects in the crystal lattice. Many recent works (Paul and Curtin, 1975, 1987; Lamartine et al., 1976; Byrn, 1982; Perrin et al., 1987) have been devoted to the study of gas-solid reactions in organic chemistry. In our own laboratory, both noncatalytic reactions-reactions of chlorine with various phenols (Lamartine et al., 1980)-and catalytic ones-reactions of isobutene on p-phenylphenol (Lamartine and Perrin, 1972) and hydrogenation of thymol (Lamartine et al., 1979; Lamartine and Perrin, 1983)-have been studied. Additional experimental work along these same lines is in progress or planned. However, very few attempts have been made to analyze the experimental results obtained with respect to the mechanisms of the reactions. In fact, many works can be found that are dedicated to gassolid mechanisms but that are concerned mainly with inorganic chemistry. In order to understand the course of an organic reaction, we have
* T o whom correspondence
should be addressed.
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examined the different models that can be found in the literature (Delmon, 1969; Levenspiel, 1972; Barret, 1973; Byrn, 1982) and selected the one that seems to be the best adapted to the experimental kinetic data that are available, for the chlorination of 4-chlorophenol.
Different Models for Gas-Solid Reactions For each of the following models, the reaction studied is assumed to be S (solid) + G (gas) initial quantities: amt transformed at time t :
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( 1 ) First- and Second-Order Kinetic Laws. In an homogeneous medium, one could say that a single-step reaction rate is proportional to the amount of reagents still available:
dx/dt = h ( a - x ) ( b - X ) 8 1989 American Chemical Society
