Ind. Eng. Chem. Fundam. 1983, 22, 209-215
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Application of Singularity Theory to Modeling of Steady-State Multiplicity: Propylene Oxidation on Platinum Moshe Sheintucht and Dan Luss’ Department of Chemlcal Engineering, University of Houston, Houston, Texas 77004
Two types of steady-state multiplicity patterns were observed during the isothermal oxidation of propylene over platinum wires. These patterns differ in the nature of the Ignition or extinction induced by slowly changing the propylene concentration. Changes in either the inert diluent, the wire temperature, or oxygen concentration caused a shift from one multiplicity pattern to the other. This information can be used to discard classes of kinetic models which cannot admit such a transition and to suggest the functional form of an appropriate rate expression. The process of model discrimination is greatly facilitated by the observation that the simplest model predicting this transition must have a pitchfork singularity. This information is utilized to develop the simplest rate expression which predicts the surprising shift in the multiplicity pattern.
Introduction Multiple steady states were observed in several isothermal catalytic reacting systems. When the reaction rate in these systems was determined as a function of an operating (bifurcation) variable, different types of multiplicity patterns (bifurcation diagrams) were obtained. For example, in the isothermal oxidation of CO on platinum (Beusch et al., 1972) and ethylene on platinum (Zhukov and Barelko, 1976) a “clockwise” hysteresis pattern was observed in which a continuous increase in the concentration of the limiting reactant led to extinction, while a continuous decrease in the concentration caused an ignition. A schematic of this bifurcation diagram is shown in Figure la. Similarly, a “counterclockwise”hysteresis was observed during the isothermal catalytic oxidation of H, over Pt (Volodin et al., 1977; Rajagopalan and Luss, 1980) and it is shown schematically in Figure lb. The knowledge of the multiplicity pattern may be exploited to determine the appropriate type of kinetic rate expression. For example, a “clockwise” hysteresis implies that the kinetic rate expression has a local maximum at some intermediate concentration and that it is single valued (rl(C) in Figure IC)or multivalued (rz(C)).On the other hand, a counterclockwise hysteresis implies the existence of an S-shaped multivalued rate expression (r3(C) in Figure Id). The existence of a single-valued rate expression with a local maximum is quite common in catalytic reactions and is usually attributed to reactant inhibition as in a bimolecular Langmuir-Hinshelwood mechanism. The catalytic reaction rate depends in general on the concentration of the limiting reactant as well as on the concentration of the various occupied surface sites, which under steady-state conditions satisfy the equation gi(c,e)= o (i = 1, 2, ...,N) where C is the concentration of the limiting reactant and 0 is a vector of N types of occupied surface sites. When the set of eq 1 has nonunique solutions for some C the reaction rate becomes a multivalued function of C. Several catalytic mechanisms that lead to a multivalued rate expression have been described by Bykov and Yablonski (1981). For example, the oxidation of CO was described by the following mechanism Technion, Kiriat Hatechnion, Haifa, Israel.
co + s Fi co-s 02 + 2s 20-s co-s + 0-s coz + 2s co + 0-s coz + s
--
which leads to a rate dependence similar to r&C) in Figure IC. Interchanging the roles of the CO and the Oz in the same mechanism gives the S-shaped rate dependence shown in Figure Id. The multiplicity pattern may be constructed in the case of a single limiting gas-phase concentration from the graphical solution of the steady-state equation kC(Cb - c) = 0) = r(C) (3) with C and 6 satisfying eq 1. When r(C) has one of the two forms shown in Figure IC, three steady-state solutions exist for a bounded range of bulk concentrations or mass transfer coefficients. Figure l e is a map of the region of mass transfer coefficients and bulk concentrations for which multiplicity exists for a kinetic expression of the form of rl. the limiting transport coefficients It,, and k,* are those for which the left-hand side of eq 3 is tangent to rl(C). It is seen that for all k , > k , a unique solution exists for all C. However, for all k, < k , multiple solutions exist over a finite range of C. The corresponding multiplicity pattern is shown in Figure la. When the rate expression is multivalued, such as rz(C) in Figure ICand r3(C)in Figure Id, there exists no more a limiting value of k, above which a unique solution exists. The region of k , and C for which multiplicity exists is described by Figure If. For all the three rate expressions described in Figure 1 the transport coefficients bounding the multiplicity regions are monotonic decreasing functions of the gas-phase concentration of the limiting reactant. The experimental determination of all the possible types of bifurcation diagrams, which describe the dependence of the steady-state rate on the concentration of the limiting reactant, is helpful in determining the qualitative features of the rate expression describing the system. In certain cases the bifurcation diagrams are either more complex than those shown in Figures l a and l b or they depend in a different fashion on the transport coefficients. The singularity theory may be used in such cases to help determine the simplest rate expression predicting these observations and for discarding rate expressions which cannot
0196-4313/83/1022-0209$01.50/00 1983 American Chemical Society
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r
Tr =240°C
Tg=25'C
O S C STATE I L
-
b'
/
-/ /"
PERT
I
I
-'b
*C,H,
CONC.(VOL %)
Figure 2. Transition from multiplicity pattern I (a, b) to I1 (c) with decreasing oxygen concentration. c, Fcb Figure 1. Schematic diagrams of observed multiplicity patterns: hysteresis in clockwise (a) and counterclockwise (b) directions, corresponding intrinsic rate functions (c, d), and domains of multiplicity (e, D. -4
predict the experimental data for any set of parameters. We report here a study of the multiplicity patterns (bifurcation diagrams) observed during the oxidation of propylene on isothermal, catalytic platinum wires. This system exhibits some novel multiplicity features which have not been reported so far in the literature and which cannot be accounted for by any rate expression of the form shown in Figure 1. We also illustrate how the singularity theory can be used to help develop a model predicting the different types of observed multiplicity patterns.
Experimental Procedure The isothermal oxidation of propylene was carried out on a 4.8 cm long, 0.005 cm diameter high-purity platinum wire (United Mineral and Chemical Co.) placed in a flow reactor and maintained at a constant resistance, and hence temperature, by a constant-temperature anemometer. A detailed description of the experimental system was presented by Zuniga and Luss (1978). Extra-dry grade oxygen (99.6% minimum purity, hydrocarbon less than 30 ppm, other impurities N P ,Ar, COP), either high-purity (99.9%) or extra-dry grade nitrogen, and high-purity (99.995%) helium and argon from gas cylinders (Linde Inc.) were passed through activated charcoal beds and then mixed with CP grade propylene by passing through a bed of Drierite pellets. The heat generated by the reaction was calculated from the difference in the electric power required to maintain the wire at a preset resistance with and without reaction. The reaction rate was computed by assuming that the reaction occurred only on the surface of the wire and that the propylene was completely oxidized to C02 and H 2 0 (AH= -462 kcal/mol). This assumption was supported by a gas chromatographicanalysis (Porapak Q column) and agrees with other studies of C& oxidation on supported platinum; Carballo and Wolf (1978) did not detect partial oxidation products at 100-130 "C and Cant and Hall (1970) found less than 2% partial oxidation products at 98 "C. The platinum wire was activated by heating it in oxygen to about 700 "C for 1h before adding propylene (2% vol)
to the mixture. After about 1h the reaction sustained itself without electrical heating and was kept at about 700 "C for an additional 4 h. The catalyst was reactivated before each run by heating in air to 800 "C for 5 min followed by heating to 500 "C for 30 min. Maximal propylene concentrations in the feed were 1.6 vol % in O2 and 2 vol % (the lower explosion limit) in mixtures containing less than 50% O2in the inert carrier. The reaction did not sustain itself under these conditions and the wire had to be heated electrically to keep it at the preset temperature. Experimental Results Two different steady-state rate branches were observed during the oxidation of propylene over an isothermal platinum wire in a certain domain of wire temperature, feed composition, and with three different inert diluents. These branches formed two distinct multiplicity patterns when the reaction rate was measured as a function of the propylene concentration. In the first pattern a high-activity monotonically increasing reaction branch existed over the whole range of the propylene concentrations employed. The wire always remained in this state following a small increase or decrease in the concentration of the propylene. Another low-activity branch existed at sufficiently high reactant concentrations. An ignition from the low- to the high-activity branch (shown by an upward arrow in Figures 2 and 3) always followed a continuous decrease in the propylene concentration. The low-activity branch was attained from the high-activity branch when the electrical current was completely turned off for several seconds causing a temporary cooling of the wire to room temperature. The electrical power required to heat the wire at the low-activity rate was identical (within experimental error) with that required for a feed which does not contain the limiting reactant. A gas chromatographic analysis did not detect any conversion at this low-activity state. Type I multiplicity pattern was observed at high oxygen concentrations (Figure 2a) and sufficiently high wire temperatures (Figure 3a, 321 "C). The ignition temperature was shifted to higher propylene Concentration with increasing O2concentration (from 0.1% at 30% O2to 0.25% at 50% O2as shown in Figure 2, and to 0.74% in undiluted 0,) and increasing wire temperature (Figure 3). Thus, ignition could be attained by increasing either the oxygen concentration or the wire temperature.
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b)
.68 v. K cp, 1.14 em/$ TI' 25.C
U.
I
t
20%4+He,
I
20X02+N,
Lf
a 5 3 70 ~ 02 + Nz , U-3 4 cm/s
05
'1 ' o b $5'
i
15 2 -CH ,, CONC (vo~.%) Figure 3. Transition from multiplicity pattern type I to type I1 with decreasing wire temperature for two inert diluents: (a) He and (b)
-
SURFACE TEMP. P C )
Figure 4. Dependence of hysteresis with respect to the wire temperature on the inert diluent and oxygen concentration (a, b) and on bulk gas temperature.
NZ. In a few experiments, close to the transition to the second multiplicity pattern, the higher activity branch was no more a monotonic function of the propylene concentration and had small local maximum and minimum points. This occurred at intermediate oxygen concentraand temperatures (Figures 3a, tion (Figure 2b, 30% 0,) 267 "C). The second multiplicity patterns consisted of a low-activity branch which existed for all the propylene concentration employed ( 0 at the singular point) and that the overall reaction is C3H6 + (6 - ~ 3 0 2 3(1 - u")CO~+ 3(1 - ~ " 3 H 2 0+ UR (15)
-
where the stochiometric coefficient u is a small number. We assume that the rate expression is of the form defined by eq 14 with KR = 0 and This quadratic dependence is assumed since a linear dependence on the product concentration fails to predict a pitchfork singularity. Substitution of (lo), (14), and (16) in (11)for the special case of CRb = 0 gives the steady-state equation
Introducing the dimensionless variables eq 17 may be rewritten as F(x,X,A,B) = A ( l - x)(l AX)' - [l
+
+ BX2(1- x ) ~ ] x=
0 (19) In the physically feasible domain of 0 I x I 1and positive A, A , and B, eq 19 has a unique pitchfork singularity at x = -1 21/2; A = 1/(8)1/2; X = 1 21/2; B = 0.5 (20) Another pitchfork singularity exists outside the feasible region at x = -1 - 2112; A = -1/(8)1/2; = 1 - 21/2; B = 0.5 (21) Balakotaiah and Luss (1981) used the bifurcation theory to divide the global realizable parameter space into regions with different bifurcation diagrams for the case of a first-order reaction in a cooled CSTR. In this problem no limit point exists on the boundaries of the feasible x , X region and the same scheme can be used to divide the realizable parameter space of A L 0 and B 1 0 into regions with different diagrams. The bifurcation theory predicts that an isola (isolated branch) can appear or disappear only on the isola variety defined by the equation
+
+
F = -aF = -aF
while the condition aFIaC, = 0 results in
213
ax
ax = o
(22)
Simiilarly, an S-type multiplicity can appear or disappear only at the hysteresis variety defined by
The isola variety for F defined by (19) is B = 4A2
(24)
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cm
-
Cpb
Figure 7. Two types of multiplicity regions for a system with
a
pitchfork singularity.
B 4 a*
I
2
s
p
5
- A
Figure 6. Bifurcation diagrams of the proposed kinetic model and the corresponding domains in the parameter plane: top figure: hifurcation diagrams at pitchfork singularity (center) four adjacent regions (corners), and boundaries between regions.
while the hysteresis variety is defined by the parametric expression
(B'+ A)'@
-3
+
~ -(B ) ~ A)A(4x3 - 62
+ 4) + A2 = 0 (25)
The isola and the hysteresis variety are shown by solid and dashed lines in the lower part of Figure 6. The two varieties divide the realizable parameter space into four regions in each of which a different bifurcation diagram exists. The upper part of Figure 6 describes bifurcation diagrams for various values of A and B. Case 0 is for the pitchfork point and any perturbation of the parameters changes this diagram to one of the surrounding graphs. Figures 6a, b, c, and d are the bifurcation diagrams corresponding to each of the four regions having the same notation in the parameter regions map (Figure 6). All the bifurcation diagrams include an isola and it can be proven that the two branches of the isola remain open even for very large values of X. The dashed branches in the bifurcation diagrams represent unstable solutions which cannot be observed. The bifurcation diagrams shown between those of any two regions are for parameters on the boundary separating these regions. In the degenerate case of B = 0 the isolas shown in regions c and d are shifted to infinity and the bifurcation diagram has an inverse S shape for all A in (0, 1/27) and of a single valued graph for all A > '/n.This degenerate case describes the usual bimolecular isothermal Langmuir-Hinshelwood kinetics whose multiplicity features were discussed extensively in the literature (Roberts and Satterfield, 1966). The bifurcation diagrams in regions a and b correspond to a multiplicity of type I, for which ignition of the low activity state can be obtained by a decrease of the concentration of the limiting reactant while the high-activity branch exists for all reactant concentrations. The bifurcation diagrams c and d describe cases with multiplicity pattern I1 in which decreasing the reactant concentration shifts the system from the high- to the low-activity branch.
The theory predids that the parameter regions, in which multiplicity patterns I and I1 (i.e., a and d in Figure 6, respectively) exist, are separated by a region in which the multiplicity pattern has three bifurcation points and are either of type b or c in Figure 6. Thus, the graphs of the ignition and extinction temperatures vs. the propylene concentration have one of the two shapes shown in Figure 7. In the first case (Figure 7a) the ignition temperature is a montonic increasing function of the propylene concentration while the graph of the extinction temperature has a local maximum at T = T,, say. A type a multiplicity pattern exists for all T > T, while for all temperatures lower than that of the cusp (To) a type d multiplicity pattern exists. A type c pattern with three bifurcation points exists for all T in (To,T,). In the second case (Figure 7b), the extinction temperature is a monotonic decreasing function of Cbwhile the graph of the ignition temperature has a local minimum at Cb= C, and T = T,. Here the temperature regions in which pattern a and b exist are separated by a temperature range (To, T,) in which pattern b exists. We have not observed in our experiments either pattern b or c. However, Lance Loban has recently observed in our laboratory an inverse S loop when he varied the concentration of the propylene in a mixture with air. Yet he was not able to find the high-concentration branch which emanates from the third bifurcation point. For a mixture of propylene in 50% O2 + N2 the ignition and extinction graphs resemble those shown in Figure 7a for C > C, (the extinction temperature was essentially unchanged for propylene concentrations in the range of (0.2, 0.33%). However, he could not find the inverse S loop, probably because it existed only over a narrow range of temperatures and concentrations. His experiments suggest that multiplicity pattern type c exists in this system, but there is a need for additional experimental verification of this point. Recent experiments in our laboratory show that a similar pitchfork behavior exists during the nonisothermal oxidation of carbon monoxide on a supported platinum catalyst (Harold and Luss, 1982). In those experiments the range of (T,,To)was rather large and could be found easily. The regions map shown in Figure 6 predicts in agreement with the experimental observations that increasing A, or equivalently increasing the mass transfer coefficient, can shift the multiplicity pattern from type I to type 11. It should be emphasized that our purpose here is mainly to illustrate how the observed bifurcation diagrams and the singularity theory can help to develop a kinetic model
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 215
describing all the qualitative features of the system, and not to get an exact kinetic model for the catalytic oxidation of propylene on platinum. However, the form of the proposed kinetic model deserves a comment. Equation 14 suggests a chain reaction mechanism. Bakaev et al. (1975) suggested that a chain reaction occurs in the gas phase following a heterogeneous initialization step in propylene oxidation on platinum. Although such a heterogeneoushomogeneous mechanism accounts for the observed pattern and rate dependence on the inert diluent, it cannot explain the constant rate obtained a t different stream temperatures (Figure 4c). Other interpretations of eq 14 are plausible, but obviously the discrimination among rival mechanisms requires a series of critical experiments to verify the various hypotheses. Concluding Remarks The main conclusion of this study is that patterns of steady-state multiplicity (bifurcation diagrams) are a very effective tool in ruling out classes of possible rate equations. Knowledge of the various possible bifurcation diagrams can be used to identify the type of singularity which can describe the observed features. This is helpful in suggesting constraints on the functional form of the steady-state equation and predicting the minimal number of parameters it needs to include. While the mathematical analysis cannot be expected to predict the physical or chemical nature of the rate processes leading to the experimental results, it is helpful in suggesting possible classes of mechanisms. When a detailed kinetic mechanism is not known as in the catalytic oxidation of propylene, we are usually satisfied to find the simplest model which is consistent with the experimental observations. The singularity theory is useful in the construction of such models and in correlating the observations. The theory enables us to reduce the arbitrariness of the mathematical representation and to select models on the grounds of simplicity and structural stability. The cubic eq 19 which includes two parameters (in addition to A) is the simplest function capable of describing a pitchfork. Thom (1975) used the singularity theory as a basis for developing speculations in several fields. It is our opinion that the singularity theory can be a very useful tool, albeit not the only one, in the discrimination among rival kinetic models based on knowledge of the various possible bifurcation diagrams. There is an obvious need to explore this suggestion further. The observed shift in the isothermal multiplicity pattern has not been reported previously for chemically reacting systems and represents a complexity not encountered before in catalytic systems. It would be of much interest to carry out additional experiments to determine how
common this complex behavior is. Acknowledgment We are thankful to V. Balakotaiah for pointing out the resemblance between the experimental results and the pitchfork singularity and for many helpful comments and discussions. This work was partially supported by the U.S.-Israel Binational Science Foundation and the Welch Foundation. Nomenclature A = dimensionless kinetic parameter, defined by (18) B = dimensionless kinetic parameter, defined by (18) C = surface concentration of limiting reactant C, = propylene concentration CR = concentration of species R k , = mass transfer coefficient ki = kinetic rate constant K = adsorption equilibrium coefficient of propylene p = vector of parameters r = reaction rate S = catalytic site IC = conversion Greek Letters a = dimensionless transport coefficient kcp/vkCg 13 = fractional surface coverage X = bifurcation parameter, defined by (18) v = stochiometric coefficient Subscripts
b = bulk phase Registry No. Propylene, 115-07-1; platinum, 7440-06-4. Literature Cited Baiakotalah, V.; Luss, D. Chem. Eng. Cbmmun. 1981, 13. 111. Bakaev, I.I.; Aseev, I.V.; Koshnayakov, A. M.; Novachenko, E. B. Zh. F k . Khlm. 1975, 4Q, 1719. Beusch, H.; Figuth, P.; Wicke, E. Chem. Ing. Tech. 1972, 44, 445. Bykov, V. I.; Yabionski, C. S. Int. J . Chem. Eng. 1981, 21, 142. Cant, N. W.; Hail, W. K. J . Catal. 1970, 16, 220. Carbailo, L. M.; Wolf, E. E. J . Catal. 1078, 53, 366. Qilmore, R. “Catastrophe Theory for Scientists and Engineers”; Wiiey: New York, 1981. Goiubitsky, M.; Schaeffer, D. Commun. Pure Appl. Math. 1979, 32, 21. Goiubltsky. M.; Schaeffer, D. I n “Nonlinear Partial Differentlal Equations in Engineering and Applied Sciences”; Sternberg, R. C.; Kaiinowski. A. J.; Papadakis, J. A.. Ed.; Marcel Dekker, New York, 1980; p 229. Harold, M.; Luss, D. AIChE Los Angeles Meeting, 1982. Rajagopalan, K.; Luss. D. J . Catal. 1980, 61, 289. Roberts, R. W.; Satterfieid, C. N. Z n d . Eng. Chem. Fundam. 1968, 5 , 317. Saunders, P. T. “An Introduction to Catastrophe Theory”; Cambridge University Press, Cambridge, 1980. Thom, R. “Structural Stability and Morphogenesls”; (English transi by D. H. Fowler) Benjamin: Reading, 1975. Volodin, Yu. E.; Bareiko, V. V.; Khal’zov, P. 1. Dokl. Akad. Nauk SSSR 1977, 234, 1108. Zhukov, S. A.; Bareiko, V. V. MI. Adad. Nauk SSSR 1978, 229, 655. Zuniga, J. E.; Luss, D. J . Catal. 1978, 53, 372.
Received for review March 1, 1982 Accepted November 17,1982
