I n d . E n g . C h e m . R e s . 1989,28, 948-954
948
Kinetics Falsification by Symmetry Breaking. 1. Steady-State Analysis Moshe Sheintuch Department of Chemical Engineering, Technion-Israel
I n s t i t u t e of Technology, Haifa 32000, Israel
Stable asymmetric states may be induced in catalytic wires controlled to attain a preset average temperature by changing the current fed into it. Complex multiplicity patterns are found there even at a fixed average temperature. A single-valued rate expression induces two bistability domains in a uniform wire. Cooling effects a t the wire edge may eliminate one of these domains. Apparent isothermal tristability may be found with a multivalued rate curve hut boundary effects destroy one of the solutions. Review of published multiplicity patterns observed with catalytic wires shows that isolated branches are due to the asymmetry. Analysis of more intricate kinetic models accounts for the bifurcation diagrams and maps observed during olefins oxidation on a Pt wire (part 2 of this work). In a recent article, Harold et al. (1987) reviewed the observations of isothermal multiplicity in catalytic systems and presented a methodology for model discrimination in bistable systems. Their analysis assumes that all steady states are space independent (homogeneous). Their review shows that the multiplicity pattern of clockwise hysteresis was observed in all studies of CO oxidation on noble-metal catalysts. Other multiplicity patterns were observed in the Pt-catalyzed oxidations of H2 (Rajagopalan and Luss, 1980; Volodin et al., 1982),of C3H6(Sheintuch and Luss, 1983), and of ammonia (Schmidt and Sheintuch, 1986). The studies cited above employed a wire whose average temperature is controlled at a preset value by changing the current. That thermochemic method is a very convenient method for kinetic research since the average reaction rate is determined from the enthalpy balance. Sheintuch and Schmidt (1986) have shown, however, that this method may induce inhomogeneous temperature profiles in which the wire is partially ignited. They applied their analysis to account for multiplicity patterns with four steady-state branches observed in ammonia oxidation. Their predictions were confirmed by Lobban et al. (1989) by a direct measurement of infrared radiation along a P t ribbon catalyzing the same reaction. This behavior is by no means unique to the thermochemic method: the local current in a Ni wire undergoing anodic dissolution under galvanostatic conditions exhibits similar behavior (Lev et al., 1988). Our previous steady-state analysis (Sheintuch and Schmidt, 1986) treated ideal systems in which the wire properties are uniform and cooling effects at the wire edges are ignored. It was also limited to systems that attain at most two homogeneous steady states at any given current. The purpose of this analysis is to expand the theory into more complex situations. (a) The first complex situation is a nonuniform wire temperature due to cooling effects at its edges. This modification allows us to account for all previous observations of multiplicity patterns on catalytic wires. (b) The second complex situation involves systems that may attain three solutions for certain currents. Studies on olefin oxidation over a P t wire showed three solutions organized in a pattern, which suggests the existence of three homogeneous steady states. These results are presented in part 2 of this work (Sheintuch et al., 1989). With these modifications, all observations of isothermal multiplicity can be accounted for by thermal effects in reactions that are inhibited by reactants. Symmetry breaking occurs when we try to achieve a surface temperature that is an unstable solution of the nonisothermal wire. The mass and enthalpy balances should be satisfied locally, at every point, in a homogeneous solution. When the desired temperature is a stable solution
of the local enthalpy balance, then a homogeneous solution is stable in the constant-temperature mode. When the local steady state is unstable, the wire reaches a global inhomogeneous state in which the wire is partially ignited (and partially extinct), where the ignited and extinguished states are the stable solutions of the nonisothermal balance. Now, is the front separating the two sections stable? In an uncontrolled wire, the front will propagate in the direction of the less stable state (to be defined later) and the inhomogeneity disappears. Control of the average temperature fixes the front in a position that satisfies the set point with a current that makes both states equally stable. The front is stable since the advance of the ignited section increases the overall resistance, resulting in an immediate decline in current, which in turn makes the extinguished state more stable. Another mechanism for fixing the front position exists when there exists a gap in heat conduction of one or more points along the wire. Such a gap can take the form of a wire neck or of a less-active nonignitable cold section (Lobban et al., 1989). The various active sections do not communicate then, and each one attains its own stable state. The front position is fixed now, and the current is adjusted to attain the overall resistance. Each homogeneous section may obtain, of course, an inhomogeneous solution of its own, which is stabilized by the previous mechanism. Both mechanisms of symmetry breaking, i.e., the fixed current with a varying front position and the fixed position with a varying current, are analyzed by Sheintuch and Schmidt (1986) and confirmed by Lobban et al. (1989). The latter mechanism was identified as the source for the two intermediate states, out of the four, observed in ammonia oxidation over a piece of Pt wire (e.g., see Appendix). We noted that the rates at these two solutions are complementary and can be explained by two mirror-image profiles. Once the upper and lower homogeneous branches were traced and the fixed front position was determined, the other branches could be predicted with good accuracy. In further experiments, we screened the wires to use those that exhibit only bistability. The bifurcation diagrams and maps observed with these wires could be accounted for by assuming that both states are homogeneous, but the necessary kinetics is quite complex. We realize now that the reactive state is an inhomogeneous solution with a varying front position at low temperatures or a homogeneous solution at higher temperatures. The transition between the two is continuous without hysteresis in contradiction with existing theory. We show that the disparity is due to the cooling effect at the wire edges, which makes the upper homogeneous state inaccessible. This conclusion is supported by quantitative comparison of theory and expesi-
0888-5885/89 12628-0948$01.50/0 C 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989 949 need to introduce heat conduction into eq 1: i.e., ( d k J 4 ) d2T/dz2 Q, - Q, = 0. We implicitly assume that the steady-state relation C ( T )is established very fast in comparison with the enthalpy balance and that diffusion is short ranged. The ideal problem assumes no-flux boundary conditions and an infinitely long wire. A front separating two sections at different steady states is stationary when the integral enthalpy balance is maintained across the front:
Q
+
Ta
-T
TS
(3) That fixes the current in a way such that areas ABS and SCD are identical (Figure la). The front position is determined by T,. The ignited fraction of the wire ( v ) satisfies 7TD (1 - 7)TA = T,; changing the temperature between TA and T Ddoes not affect the current but does change the front position. The transition back into the homogeneous solutions occurs at T, = TA or T, = TDas the front reaches the edge. Thus, the temperature should be continuous at these transitions with discontinuity of the slope-a feature that is unique to symmetry breaking. This analysis assumes a very narrow front, in comparison with the wire length and the absence of end effects. These assumptions are invalid when the front rests near the edges, as we show later. By varying the reactant concentration or another parameters, we can follow the changes in the positions of transition points A-D and plot their loci in the parameter plane (Figure lb). All the bifurcation points coalesce at the boundary from uniqueness to multiplicity, Le., when Q, and Q, are tangent at the inflection point. That is the cusp point of the nonisothermal problem (point a). The isothermal rate curve at T, < T , (or T, > T,) is described by line 1 (or 2 ) in Figure 3: it should exhibit a kinetic branch for C < CB (or C < Cc) with a bifurcation at C A (or C D ) to higher (or lower) rates along the inhomogeneous branch. That is the pattern observed during H2 oxidation by Rajagopalan and Luss (1980). The nature of the isothermal rate curve is determined by the inclination in the (T,Cb)plane of the boundaries corresponding to transition points A, B, etc. The disposition shown in Figure l b corresponds to positive-order reactions. When the reaction rate is nonmonotonic, the ignition line ( T , = T B )may change direction and ascend with c b . A t T , = TB, we should satisfy dQ,(T,Cb)/dT = h. From a second derivation (d(dQ,/dT)/dCb= 0), we find
+
cb Figure 1. Bifurcation diagrams (a, c) and maps (b, d) for reactions of single-valued rates catalyzed by wires without (a, b) or with (c, d) end effects. Map d accounts for nonmonotonic kinetics. (Thick lines denote observable branches or boundaries, thin solid lines are unobserved branches, and broken lines are saddle points.)
ments. The required kinetics can then be simplified considerably.
Analysis This section contains the following hierarchy of steps: (a) reviewing the ideal situation of a uniform wire catalyzing a reaction characterized by a single-valued (i.e., monotonic kinetics) sigmoidal heat-generation curve; (b) refining the theory for situations where end effects are important; (c) repeating the analysis for kinetics that induce isothermal bistability; (d) extending the results to a heat-generation curve that can sustain three stable steady states. For each problem, we construct the bifurcation diagram observed when the controlled-wire temperature (T,) is swept at various concentrations. We construct, then, the bifurcation map in the temperature versus concentration plane. From its cross sections, the rate dependence on reactant concentration can be constructed. (a) Basic Ideal Problem. The local enthalpy balance accounts for heat generation by reaction, Joule heating, and heat loss at steady state 12R Q,(T) + - - h(T - T f )= 0 nd
The heat-generation curve Q, = (-AH)r(C,T)already incorporates the steady-state relation between temperature and concentration obtained from the isothermal balance k,(Cb - c) = r(C,T). The steady state is the intersection of the nonlinear Q,(T)and the linear Q,(T),which can be rewritten in the form
Changing the current amounts to parallel displacement of Q, (changing Tb)accompanied by small changes in its slope due to the temperature effect on the resistivity: R = R,[ 1 + a(T - T f ) ] .When Q, is single valued but steep, as in Figure l a , the surface temperature along section BC is unstable: B and C are the tangency points of Q, and Q,. Setting the average temperature at TB < T, < Tc leads to symmetry breaking. To find the global steady state, we
dTB - _ - d2Qg/dTdCb dCb
d2Q,/dT2
(4)
The slope changes direction, therefore, when the reaction is of apparent zero order (dQ,/dCb = 0). The slope of the extinction ( T , = TA) line can be determined from the concentration derivatives of the stationary front condition (eq 3) and of the steady-state function (eq 1). For the case of a = 0 (eq 2), the slope is described by
The heat generation at the lower steady state is usually small and only weakly dependent on temperature or concentration. The slope of the transition line (TA) is de-
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Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
termined mainly by the average value of dQg/dCb. For positive-order reactions, TAdeclines with c b . For nonmonotonic kinetics, the extinction line may be of positive or negative slope. At low temperatures, dQ,/dcb < 0, while when the upper steady state is mass transfer controlled the high-temperature contribution is positive. (b) Wire-Edge Effects. The ideal problem is characterized by the similarity of symmetry breaking from the lower or upper homogeneous state. The cooling effect a t the edges tends to stabilize the lower state and to destabilize the upper one: compare the half profiles plotted in Figure 1, a and c. The former is that of a uniform wire with an extremely narrow front showing zero heat flux a t the edges. A more realistic condition is that the wire-edge temperature reaches the fluid temperature ( Tf). Assuming symmetry at the center, the problem is described now by
When Tfis below the lowest steady state, the ignited state looks like the inset in Figure IC. There exists now a front near the edge. Its width depends on the operating conditions and the fluid temperature. Now, can that state be established below the current that maintains the asymmetric state? Multiplicity of solutions cannot be ruled out then since the center temperature is not specified. However, for sufficiently long wire, T(0) T,, and the domain of attraction of that state becomes extremely narrow. Thus, as the temperature is decreased along the upper branch, part or all of section CD will not be established since the front already exists in the system. Close to point D the front will move away from the edge and toward the center as the temperature declines. The observed rate is continuous with discontinuity in the derivative (Le., supercritical bifurcation), unlike the (subcritical) bifurcation encountered before. Now, since the front and the wire are of finite width, the end effects become pronounced as the front reaches the edge. The transition to homogeneity will be premature (at T , > T A )and will appear as a regular extinction. The distance between the two branches is expected to be small. Increasing the temperature from the lower branch yields the same sequence of states as in the ideal problem. We portrayed in Figure 1 only half-profiles since with increasing T, the ignition at B will be initiated at the center, which is the hottest point, and a symmetric inhomogeneous profile will be formed. The map in Figure Id accounts for the two independent effects: the ignition and extinction lines ( TAand T B ) acquire a local minimum due to reactant inhibition. The Tc line is not observable due to end effects, and the TD line marks the exchange of stability from the homogeneous to inhomogeneous solutions. The rate curve obtained by a cross section of the map (shown in Figure Id) is continuous for T , > T , (line 3, Figure 3) and is discontinuous at T, < T,, showing either two hysteresis loops (i.e., mushroom shaped, line 4) or an isolated branch (isola, line 5 ) . It is single valued a t sufficiently low temperatures: These patterns were observed during isobutylene oxidation and are presented in part 2 of this work (Sheintuch et al., 1989). The bifurcation diagram with varying temperature for the nonuniform wire shows only one hysteresis loop (Figure IC) and is clearly identical with the observations made during the oxidation of C3H6(Figure 2) and NH3 (see Appendix). The slope of the upper branch is similar to the heat-transfer coefficient; Le., this branch is parallel to the line of heat loss in the absence of reaction. Changing
300 h
0 v
I-"
250
l a
I
200
.5
.25 X&%V)
7 v YO C3H6 in 0, U=248cm/s
i
-
200
300 -A
200
400
-
I I
300
C 400
SURFACE TEMP ("C)
Figure 2. Bifurcation map (a) observed during atmospheric oxidation of propylene on a Pt wire, and bifurcation diagrams showing the effect of diluent (b) and of gas temperature (T,) (e) (after Sheintuch and Luss (1983)).
5
I
-
I
cb Figure 3. Bifurcation diagrams showing the change in rate with reactant concentration (notation as in Figure 1, the diagrams correspond to marked cross sections of maps in Figures 1 and 4).
the diluent from N2 to He changes the slope of the upper branch and affects the location of the ignition and extinction points, as expected (Figure 2b). Increasing the fluid temperature (Tf)compensates for part of the Joule
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 951
cb Figure 4. Bifurcation diagrams (a, c) and maps (b, d) for reactions with multivalued rate catalyzed by a wire without (a, b) or with (c, d ) end effects (notation as in Figure 1).
heating (lower Tb, eq 2), but it does not affect the diagrm (Figure 2c). (c) Multivalued Heat-Generation Curve. Reactant inhibition will lead to isothermal multiplicity at sufficiently high concentrations. That implies a multivalued Q, (Figure 4a) and implies that in order to determine the condition of a stationary front we need to solve the diffusion-reaction equation simultaneously with the conduction-heat-generation problem. In many situations, the diffusion front is much narrower than the conduction front. The problem can be formulated then into an inner isothermal diffusion front (at T*) with an outer conduction front (Sheintuch, 1987). The position of the diffusion front (at T*) is determined from the condition for an isothermal stationary concentration front, which is determined in analogy to eq 3
To find the sensitivity of T * to bulk concentration, we apply the derivative d / d T * to eq 7 . For a rate expression of the form r = k(T*)f(C), we find
Once T * is known, the stationary thermal front can either be determined from eq 3 or be determined graphically, when integrating along the lower branch for T A < T, < T* and along the upper branch from there on (Figure 4a). The slope of the extinction line can also be determined, in similarity to eq 5. Since Q, along the lower steady state is small, there are two contributions to dTA/dCb: a negative one from dQ,/aCb averaged on the upper branch and a positive one from the adjusting isothermal concentration front (eq 8). The slope may acquire, therefore, a positive value or a negative value. We show later than at large Cb the slope is negative. We plot now the bifurcation map for the ideal case (Figure 4b): It exhibits the familiar transition to inhomogeneity around point a, where Q,(T) is still single valued. With increasing concentration, the heat generation becomes multivalued and TBshould intersect Tc as they enter the domain of isothermal multiplicity. These lines bound now the region of tristability where two homogeneous solutions and one inhomogeneous solution are possible.
Introducing the cooling effect of the edges eliminates the tristability, and only one or two solutions is possible. The bifurcation diagram and map (Figure 4, c and d) are transformed in the way shown before for single-valued rates. The premature extinction will also tend to shift the observable cusp point away from point a. The resulting bifurcation map, when isothermal multiplicity is possible, is cusp shaped with ignition and extinction lines of opposite inclination, in similarity to the observed maps of C3H6 (Figure 2) and NH3 oxidation (Appendix). The temperature of ignition from the homogeneous lower state ascends with Cb due to reactant inhibition. We show below that the extinction temperature, i.e., the transition from the inhomogeneous state to homogeneous state (close to TA, Figure 4c), is likely to decline with c b . The corresponding sequence of bifurcation diagrams is shown in Figure 3 (lines 6-8). To show that TA declines with cb, consider the case where the upper branch is mass transfer controlled; Q, is then approximated by a step function: Q, = 0 at T, < T * and Q, = hAT at T > T * where A T is the adiabatic temperature rise. We immediately find TA = T * - AT/2. The rate is Q = h(T, - T * AT/2) T * - AT/2 < T, < T * + AT/2
+
Q = hAT
T*
+ AT/2
< T,
(9)
The "isothermal" rate curve along the upper (inhomogeneous) branch is linear with Cb when T * varies slowly with c b , in agreement with observations. The slope of the T A line is d(T* - AT/2)/dCb, and at large concentrations, the main contribution is due to the adiabatic temperature rise: dAT/dCb is O(lO0 'C/l vol %), while from eq 8 with an Arrhenius temperature dependence, we find d T */dCb R(T*)'/ECb. Thus, TA should decline linearly with concentrations at large c b . These ideas are tested in the Appendix by analyzing an experimental example of a wire, with a fixed-position front, catalyzing ammonia oxidation. The bifurcation diagram with varying temperatures in the case of isothermal multiplicity is similar to that when the rate is unique (Figures ICand 4c) since the domain of overlapping rates is eliminated. The discrimination between the two is important for modeling purposes, and we suggest now a quantitative test for that purpose. If we approximate Q, in a piecewise linear fashion, then the ignition temperature occurs in the first half of domain A-D (i.e., 2TB< TA TD)when Q, is single-valued and in the second half or even outside this domain (TB> T,) when Q, is multivalued. (The latter case is demonstrated in Figure 4 (Sheintuch and Schmidt, 1986)). The results portrayed in Figure 4 apply only when the front is stable. Front instability may result from complex kinetics or when the diffusion front is too wide in comparison with the thermal front. When the front is unstable, the bifurcation set is described by the homogeneous ignition and extinction points (B and C). With reactantinhibition kinetics, both transition temperatures ascend with concentration. (d) Homogeneous Tristability. We turn now to systems that attain up to five solutions to the local enthalpy balance. The isothermal rate expression (and Q,) may be single valued, but Q shows two inflection points as in the case of two first-order consecutive reactions. There are two independent cusp points now (al and a2 in Figure 5 , map I) that generate two independent fronts (AD and EH in Figure 5 , diagram a) with different currents. With increasing concentration, we enter the domain of three
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952 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
0
‘b Figure 6. Bifurcation diagrams (a-d) and map (I) for reactions of multivalued rates that sustain five homogeneous steady states; end effects are incorporated. Figure 5. Bifurcation diagrams (a-d) and maps (I, 11) for reactions that may sustain five (three stable) homogeneous steady states. The rate is single valued here, and end effects are absent in I and are incorporated in 11.
overlapping solutions, and for certain cb’s, there exists a current that sustains either front or a large front between A and H (diagram b of Figure 5). This is the inception point of the large front which exists for higher concentrations. The small fronts are eliminated when their local steady state undergoes ignition or extinction to another branch. Diagram c of Figure 5 shows such a point at the coealescence of states E and C. In the bifurcation map, the singular point c occurs at the intersection of the C and E lines. Finally, at point d, the intermediate branch (section CF) disappears. That appears as a local extremum in the bifurcation map. For larger concentrations, the situation is identical with the case of simpler kinetics shown in Figure 4. The nonuniformity of an actual wire “kills” the ignited homogeneous states when an inhomogeneous state coexists (branches CD and GH in diagram a of Figure 5). With changing concentrations, the currents that sustain the two fronts intersect (diagram b of Figure 5b). Beyond this point, the large front is established and the upper small front is unaccessible. The lower small front may still be stable and accessible, and i t disappears at point d. The corresponding bifurcation map is shown in Figure 5, map 11. Reaction inhibited by a reactant and characterized by a high activation energy may admit three stable solutions to the local enthalpy balance. This was verified analytically (Tsotsis et al., 1982) as well as in many oxidation reactions on a porous catalyst (Harold and Luss, 1985; Sheintuch and Avichai, 1988). A sequence of Qg’sthat admit isothermal multiplicity at low temperatures is shown in Figure 6. Various scenarios may be envisioned now: let us assume that the the inactive and intermediate states are stable while the small front between the two is unstable. At low concentrations, therefore, the bistability domain is bounded by the positive-sloped T, and Tc lines. A t higher concentrations, the other small front is formed and evolves in similarity to the behavior shown earlier (end effects are incorporated): at point c, the front meets an extinguished intermediate state and at d the intermediate state disappears altogether. At higher concentrations, the large front must exist, in similarity to the situation shown
in Figure 4. The transition between the two fronts should be discontinuous in the ideal case. We expect that in an actual system, due to the infinite width of the front at TC and due to wire end effects, the big front will be formed by a continuous transition from the small one. The resulting bifurcation map, shown in Figure 6, map I, was observed in ethylene and propylene oxidations.
Discussion We consider the erroneous kinetic implications of the bifurcation sets presented here when using a methodology that assumes that all states are homogeneous. If the steady-state model of the system is described by the algebraic equation F(X;Cb ,Cb2,T,,p) = 0, then these kinetic implications are derived from two types of mathematical conditions: The dependence of F on the variable (say x, conversion) should become more nonlinear as more solutions are observed. To account for bistability (Le., two stable solutions and one unstable solution), the model should admit a cusp point in the parameter plane defined by F = F, = F,, = 0; i.e., a cubic equation can describe the behavior in the vicinity of this singularity. In fact, the observed cusp points in the thermochemic method, such as point a in Figures 1 and 4-6, are cusps of the nonisothermal balance G(x,T(x);cb,,Cb,,p) = 0, which satisfy G = G, = G,, = 0. When tristability exists, as in Figure 6, then three cusp points are usually possible in the parameter plane and there also exists a point in the parameter space when the four derivatives of F vanish. In fact, that condition should be applied to the nonisothermal balance G, and the bifurcation maps in the plane may acquire only two cusp points (Figure 6). The dependence of F on a parameter (cblor cb2)becomes more complex as the bifurcation map acquires more extremum points. These are isola points at which either two branches coalesce or a closed loop of solutions (isola) shrinks into a point and disappears. The defining condition of such a point is F = dFfdx = FIdA = 0, where A is the direction that exhibits the isola point. A local extremum with varying A = Cb in the thermochemic method is either an extremum of the nonisothermal ignition line (i,e,, G = G, = GA= 0), induced by reactant inhibition, or a local extremum of the extinction line induced by increasing exothermicity with cb. A local extremum in the temperature direction (diagram d in Figures 5 and 6) implies a transition to negative temperature dependence of
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 953 1
I
1
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r1
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NH 3 OXIDATION
I
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/ PT WIRE
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FOUR STABLE
T P , T; ~~
I
I
1
2
, Til
3
NH3 CONC. ( V O L 'A 1 Figure 7. Bifurcation diagram and map (a, b) observed during atmospheric oxidation of NHBon a nonuniform Pt wire (after Sheintuch and Schmidt (1986)). Qualitative temperature profiles along branches I (c) and I1 (d) and along Q, (extinction) (e, f) are shown.
we will show that oscillations in the constant-temperature mode cannot be used as a proof for the kinetic source of the behavior.
Acknowledgment This work was supported by the Israel Academy of Science and Humanities.
1 TS
I
cb Figure 8. Bifurcation diagram (a) and map (b) for a wire composed of two uncommunicating sections.
the rate in truly isothermal systems. Actually this is a cusp point of the nonisothermal balance (G = 0).
Conclusions We have shown that asymmetric states lead to surprisingly complex multiplicity features even in systems with relatively simple kinetics. Analysis of such diagrams that assumes homogeneity of all states results in significant falsification of kinetics. Our review of previous work and the work presented in the second part of this work (Sheintuch et al., 1989) show that clockwise hysteresis is probably the only truly isothermal pattern that has been observed with varying reactant concentrations. The thermochemic method has been extensively used to study isothermal oscillations. In a future publication,
Nomenclature cb, c = bulk and surface concentrations d = wire diameter F , G = steady-state functions f = rate function h, k , = heat- and mass-transfer coefficients, respectively k , = heat conductivity k = rate constant I = current n = apparent reaction order Qg,Q, = heat-generation and heat-removal functions p = vector of parameters Rf,R = wire resistances at the fluid temperature and at the steady state r = reaction rate T , = preset average temperature T , Tf= surface and fluid temperatures, respectively ?'b = equivalent ambient temperature T* = temperature of the diffusion front (eq 6) x = state variable (e.g., conversion) z = length coordinate Greek Letters cy = resistance-temperature slope h = bifurcation variable q = ignited fraction of adjustable front ?* = fraction of the wire in the fixed-position front Subscripts B, E, a, c, etc. = at points B, E, a, c, etc. x , h = partial derivative with respect to x, h
954 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
Appendix Fixed-Position Front Application to Ammonia Oxidation. In a wire composed of two uncommunicating sections (of fractions q* and 1 - q * ) , two inhomogeneous states may exist as well as the two homogeneous ones. These four branches were observed during ammonia oxidation (Figure 7 ) . In our previous analysis, we treated the two intermediate branches as completely homogeneous. The rate along the intermediate branches was determined by taking the average of the two homogeneous states that lie a t the intersection of the heat-removal line with the lower and upper branches. This implies that ignition from all branches occurs along the same heat-removal line and that the distances between the inactive branch 0 and branch I is about the same as that between I1 and 111. These predictions agree with the observations made during ammonia oxidation (Figure 7 ) . The theory predicts, however, that these properties apply for extinction points as well (see Figure 4 in Sheintuch and Schmidt (1986)). We could not detect any individual extinction points but rather a continuous decline of heat generation a t a slope of the heat-transfer coefficient (Figure 7 ) . Now that we realize the the cooling effects a t the edges favor the existence of inhomogeneous solutions over homogeneous ones, when the two coexist, we can account for the extinction process. Along the line that satisfies the integral condition, the ignited section(s) is inhomogeneous, and the front shifts as the temperature declines. Declining from branch 111 yields two inhomogeneous uncommunicating sections, each with an average temperature of T,, and the front position is determined as before. Decreasing the temperature from branch I1 to branch I yields an extinguished section ( Q 0), and thus the average temperature of the ignited section is T J ( 1 - q*) and T s / q * respectively. , The front position adjusts itself to these averages. Now, since the fronts are very narrow, the overall ignited fraction of the wire must be identical in all three branches and so must the rate. The transition from these three branches to the inactive state occurs simultaneously and prematurely as the front reaches the edge. The Q, curve is multivalued in this case, as can be verified from the location of the first ignition point (Figure 7 ) . We can test, therefore, the stepwise approximation for Q, (eq 9). To test this construction, we have replotted the upper rate curve observed during ammonia oxidation with 20% 02:Originally, it was drawn versus ammonia concentration on fixed T, (Figure 7 of Sheintuch and Schmidt (1986)) and showed linear dependence. By replotting it versus temperature, we found that T * = 120 "C for C b > 2 vol % and it increases slowly at lower concentrations (123 a t 1.5%, 125 a t 1%). The rate along the upper branches is described very well by eq 9 with h = 0.013 cal/(s cm2
-
K) and AT = 50 "C/vol % NH3. We are now ready to draw the bifurcation map. Using the stepwise approximation, we can now add the intermediate branches to the Q, versus T diagrams (Figure 8a) using the q* value that was determined experimentally. The ignition points from those branches are given by T E - TB = q*AT and T , - T B = (1 - q*)AT. The extinction points of all active branches coincide; the slope of their loci should be -AT/2 a t large Cb. The bifurcation map takes the form shown in Figure 8b, in complete qualitative agreement and good quantitative agreement with the experimental map (Figure 7 ) . Literature Cited Harold, M. P.; Luss, D. An Experimental Steady-State Multiplicity Features of Two Parallel Catalytic Reactions. Chem. Eng. Sci. 1985, 40, 39-52. Harold, M. P.; Sheintuch, M.; Luss, D. Analysis and Modeling of Multiplicity Features. 2. Isothermal Experiments. Ind. Eng. Chem. Res. 1987,26, 794. Lev, 0.;Sheintuch, M.; Pismen, L. M.; Yarnitski, H. Standing and Propagating Wave Oscillations in the Anodic Dissolution of Nickel. Nature 1988, 38, 458. Lobban, L.; Phillippou, G.; Luss, D. Standing Temperature Waves in Electrically Heated Catalytic Ribbons. J . Phys. Chem. 1989. 93, 733. Rajagopalan, K.; Luss, D. Influence of Inerts on Kinetic Oscillations during the Isothermal Oxidation of Hydrogen on Platinum Wires. J . Catal. 1980, 61, 289-309. 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. Sheintuch, M. The Determination of Global Solutions from Local Ones in Catalytic Systems Showing Steady State Multiplicity. Chem. Eng. Sci. 1987,42, 2103-2114. 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. Sheintuch, M.; Schmidt, J. Observable Multiplicity Features of Inhomogeneous Solutions Measured by the Thermochemic Method: Theory and Experiments. Chem. Eng. Commun. 1986,44,33-52. Sheintuch, M.; Schmidt, J.; Rosenberg, S. Kinetics Falsification by Symmetry Breaking. 2. Olefin Oxidation on a Platinum Wire. I n d . Eng. Chem. Res. 1989, following paper in this issue. Tsotsis, T. T.; Haderi, A. E.; Schmitz, R. A. Exact Uniqueness and Multiplicity Criteria for a Class of Lumped Reaction Systems. Chem. Eng. Sci. 1982, 37, 1235-1243. Volodin, E'. E.; Barelko, V. V.; Khalzov, P. I. Investigation of Instability of Oxidation of H2 and NH3+H2 Platinum. Chem. Eng. Commun. 1982,18, 271-285.
Received for review May 2, 1988 Revised manuscript received January 24, 1989 Accepted February 22, 1989
