O2 Mixtures on Platinum: Ignition

3170

J. Phys. Chem. B 1999, 103, 3170-3178

Catalytic Ignition of H2/O2 Mixtures on Platinum: Ignition Temperatures and Postignition Transients D. Kulginov, M. Rinnemo, and B. Kasemo* Department of Applied Physics, Chalmers UniVersity of Technology, S-412 96, Go¨ teborg, Sweden ReceiVed: July 14, 1998

We report experimental and theoretical studies of the transient behavior of the catalyst temperature during ignition of H2/O2 mixtures on a platinum catalyst. The numerical simulations are based on a kinetic scheme of the reaction and a linear model of mass and heat transfer in the gas phase. We find a good agreement between experiment and theory. The influence of kinetic phase transitions on the ignition process is discussed.

1. Introduction Catalytic ignition is the sudden self-acceleration of an exothermic, heterogeneous reaction, occurring at a critical temperature where the release of chemical energy cannot be balanced by heat losses in the system.1 The physics behind ignition is as follows: As the temperature is increased, the rate of the exothermic reaction increases faster than the ability of the system to dissipate heat, and beyond some point (the ignition temperature) a steady-state cannot be sustained anymore. The system then ignites, passes through a transient regime, and arrives to a new steady-state regime where the reaction rate is controlled by mass transport rather than by the reaction kinetics. The quantitative condition for ignition, as given by FrankKamenetskii,1 is

dQc/dT >1 dQ1/dT

(1.1)

where Qc and Ql are the temperature-dependent chemical power release and heat loss function, respectively. Usually Qc is nearly exponential in temperature (Arrhenius behavior), while Ql is nearly linear in T below the radiation loss temperatures. Catalytic ignition is interesting from a purely basic point of view as a critical phenomenon. It sometimes involves sudden qualitative changes in the reaction kinetics, primarily kinetic phase transitions (for a review, see ref 2). Catalytic ignition is also important from a practical point of view, since it is related to the so-called light-off phenomenon in catalytic converters for car emission cleaning. A large number of experimental and theoretical studies of different aspects of catalytic ignition have been performed in the past. They include catalytic heterogeneous reactions with possible multiple steady states, the transport of the reactants to the catalyst surface, and the heat dissipation by heat conductance and convection.3-17 In many of these studies the catalytic reaction was represented by a simple Arrhenius model with an effective (or apparent) activation energy without explicit treatment of the kinetics3,4,10 or the catalyst surface was just regarded as a source or sink of certain bulk species.11 However, it was early pointed out6 that the Arrhenius type, one-step reaction is not sufficient to explain the observed dependences of ignition temperatures on reactant mixture compositions. More recent studies7,8,12-17,22 involve direct (explicit) modeling of the surface kinetics. The modeling of mass and heat transport has varied

from simple linear phenomenological relations5,7-9 to solving the fluid dynamics equations and taking into account gas-phase reactions.4,10,11,14-16,22 While the ignition point has received considerable attention, much less has been done on the transient behavior of the igniting system after the ignition point has been passed. (The quantities of interest in this regime are the temperature, the surface and gas-phase concentrations of reactants, and the reaction rate.) The theoretical description of the very detailed experimental study in ref 7 lacked realistic account of mass and heat transport and was not able to explain, e.g., the gradual temperature increase and temperature overshoot after ignition. The relationship between ignition and kinetic phase transitions also remains unclear. Sometimes, the postignition state demonstrates complex behavior like spontaneous extinction and even periodic and nonperiodic oscillations of temperature and reaction rate (see, e.g., ref 18). Such regimes have not been studied in detail either. In this paper we explore experimentally and theoretically the transient behavior of the catalyst temperature during catalytic ignition both at zero and finite flow of the reactant mixture. We analyze the factors that govern the shapes of the transient temperature curves (ignition traces) and the postignition steady states. We also address the role of kinetic phase transitions in the course of ignition and discuss different possible sequences of kinetic events, including extinction and reversible kinetic phase transitions. The composition of the paper is as follows: The experimental setup is briefly discussed in section 2. The models of reaction kinetics and mass and heat transfer are discussed in section 3 together with the numerical procedure employed. The results on ignition temperature and ignition traces are discussed in section 4 as are the role of kinetic phase transitions and the conditions for spontaneous extinction. 2. Experiments The experimental setup was the same as in our previous work on catalytic ignition in H2 + O2 and CO + O2 mixtures over Pt.12,17 In short, it consisted of a 0.127 mm thick and 17 mm long polycrystalline platinum wire (Material Research Corp., 99.995% purity) mounted in a quartz tube flow reactor of 19 mm inner diameter. The wire was mounted diametrically in a four-point probe arrangement to allow simultaneous resistive heating and measurement of the temperature of the wire via its resistivity (see Figure 1 in ref 12).

10.1021/jp9830068 CCC: $18.00 © 1999 American Chemical Society Published on Web 04/03/1999

Catalytic Ignition of H2/O2 Mixtures on Platinum

J. Phys. Chem. B, Vol. 103, No. 16, 1999 3171

TABLE 1 sticking coeffs of hydrogen and oxygen at low coverage: SH2(0) ) 0.05; SO2(0) ) 0.02 impingement rates of hydrogen and oxygen at T ) 300 K: FH2 ) 1.08 × 1021 s-1 mbar-1 cm-2; FO2 ) 0.28 × 1021 s-1 mbar-1 cm-2 preexponential factor for desorption of hydrogen: ν ) 1013 s-1 activation energy for desorption of hydrogen, eq 3.5: E0d ) 19 kcal/mol; A ) 7 kcal/mol diffusion coeffients of hydrogen and oxygen in nitrogen: DH2 ) 1.9 cm2 s-1; DO2 ) 0.5 cm2 s-1 diffusion coeffients of hydrogen and oxygen in helium: DH2 ) 2.5 cm2 s-1; DO2 ) 1.2 cm2 s-1 thermal conductivity and diffusivity in nitrogen: λ ) 0.5 mW cm-1 K-1; κ ) 0.7 cm2 s-1 thermal conductivity and diffusivity in helium: λ ) 1.7 mW cm-1 K-1; κ ) 2.7 cm2 s-1 radius of the wire: r0 ) 0.006 35 cm heat capacity of the wire: C ) 0.38 mJ K-1 cm-1 reaction exothermicity: ∆H ) 4 × 10-18 J emissivity of the wire: s ) 0.2 coeffients in the temp dependence of the resistivity of the wire: F0 ) 0.09 Ω cm-1; a ) 3.53 × 10-3 K-1; b ) -5.25 × 10-7 K-2 external radii for concns and the temp, respectively: Rn ) 0.5 cm; RT ) 0.95 cm geometry corr factor, eq 3.12: γ ) 2/3

The gas mixture of H2 and O2 at a total joint partial pressure of 45 Torr in nitrogen or helium was fed into the tube at atmospheric pressure with a net flow in the range 0-300 cm3 min-1. The wire was cleaned before each measurement for 2 min at 1350 K in a reactant mixture with R ) 0.15, where R ) PH2/(PH2 + PO2) is the relative concentration of hydrogen. After cooling of the system by a flow of pure nitrogen, a desired reactant mixture was introduced and the temperature of the wire was increased stepwise by increasing the electric current through it. After each step, the system was given time (5-10 s) to stabilize. The ignition eventually appeared as a spontaneous temperature increase after the electric current step, without further increase of electric current. The ignition temperature was measured for mixtures in the range 0.1 < R < 0.8, and the temperature versus time curves (ignition traces) were recorded. 3. Model A. Kinetic Model. To simulate the reaction kinetics, we employ the kinetic model developed by Hellsing and co-workers (HKZ).19 This model is based on UHV and TPD data and has been shown to reproduce the experimentally observed ignition temperatures.12,13 The main assumptions of the model are (i) adsorption of hydrogen and oxygen is dissociative and competitive, (ii) desorption of oxygen atoms and molecules and hydroxyl molecules is negligible, and (iii) the reaction rate constants are much larger than the adsorption rate constants. Since the time scale of the surface reaction is much shorter than the characteristic overall ignition time, the steady-state approximation can be used to calculate the reaction rate, and the full kinetic equations may therefore be simplified.12,19 Under such assumptions, in the steady state, one of the coverages always dominates: θH , θO ≈ θ or θO , θH ≈ θ with θ ) θH + θO being the total coverage and H and O referring to hydrogen and oxygen atoms. In the former case, when the surface is predominantly covered with oxygen, hydrogen desorption is negligible and all the adsorbed hydrogen is converted to water as well as all the adsorbed oxygen; i.e., the rates of H2 and O2 adsorption are related as

FH2SH2(θ) ) 2FO2SO2(θ)

(3.1)

where FH2 and FO2 are the impingement rates and SH2 and SO2 are the sticking coefficients for hydrogen and oxygen, respectively. (Oxygen desorption is negligible due to the low temperature and high adsorption energy of O on Pt.) In the case of hydrogen excess, we also have to take into account the second-order desorption of hydrogen. The balance equation therefore deviates from (3.1):

FH2SH2(θ) ) 2FO2SO2(θ) + kHd 2θ2

(3.2)

Following refs 12 and 19, we employ the simplest powerlaw expressions describing the dependence of oxygen and hydrogen sticking coefficients on the total coverage, θ ) θO + θH,

SO2(θ) ) SO2(0)(1 - θ)3

(3.3)

SH2(θ) ) SH2(0)(1 - θ)1/2

(3.4)

and the mean-field expression for the rate constant of hydrogen desorption,

kHd 2(θ) ) ν exp[-(E0d - Aθ)/kBT]

(3.5)

The values of the constants in eqs 3.1-3.5 (Table 1) are based on UHV adsorption and TPD data.19,12 The exponents in eqs 3.3 and 3.4 are chosen on the basis of available experimental data on H2 and O2 adsorption to reproduce the observed dependence of the ignition temperature on the mixture composition.12 This choice is not crucial (for example, second-order adsorption of O2 is almost as good), but it is absolutely necessary that the exponent for O2 is considerably larger than that for H2. The steady-state mean-field eqs 3.1 and 3.2 can be solved numerically and have one to three solutions depending on the value of R. A single steady-state solution corresponds to the surface mainly covered by either oxygen or hydrogen. In the case of three solutions, one of them is not stable with respect to local perturbations of coverage2 (hence we do not consider it), while the other two are stable and correspond to an oxygenor hydrogen-covered surface, respectively. The water production rate is calculated in each case as

W ) 2FO2SO2(θ)

(3.6)

If more than one solution exists, the kinetics are bistable and the exact state of the system is determined by its history, e.g., by the initial conditions. Slow changes of the partial pressures of the reactants can cause an abrupt transition from one stable steady state to another (hydrogen-covered to oxygen-covered surface or vice versa), so-called kinetic phase transitions.2 During these transitions, the reaction rate can change significantly. One of the purposes of this work is to investigate the relationship between ignition and kinetic phase transitions in and near the regime of bistability. B. Heat and Mass Transport. Modeling heat and mass transport in the system, we follow a slightly modified approach compared to the papers.12,17 Since the temperature and density

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variations along the wire are much smaller than those in the perpendicular direction, we identify cylindrical domains around the wire where the temperature and density fields have approximate cylindrical symmetry. The radii of the cylinders may differ for temperature and for the two reactants because of the difference in the corresponding transport coefficients. They are also likely to depend on the velocity of the gas mixture.12 In these domains, we use 1D linear diffusion and heat conductance equations

∂ni 1 ∂ ∂ni ) Di r i ) H2, O2 ∂t r ∂r ∂r

(3.7)

∂T 1 ∂ ∂T )κ r ∂t r ∂r ∂r

(3.8)

for numerical densities of the reactants ni and temperature T, where Di are the diffusion coefficients of the reactants in the carrier gas and κ ) λ/(FgCP) is the thermal diffusivity (with λ being the thermal conductivity, Fg the density, and CP the heat capacity of the carrier gas at constant pressure). At the surface of the wire, the boundary conditions for the concentrations are given by

DH2 DO2

∂nH2 ∂r ∂nO2 ∂r

)W

(3.9)

W 2

(3.10)

)

where W ) W(nH2,nH2,T) is the reaction rate dependent on the conditions close to the surface of the wire according to the eqs 3.1-3.6. The corresponding boundary condition for the temperature can be written as

∂T ∂T ) 2πr0 W∆H + λ - sσT4 + I2F(T) (3.11) ∂t ∂r

C

(

)

where C is the heat capacity of the wire per unit length, r0 its radius, ∆H the reaction exothermicity, s the emissivity of the wire, σ the Stefan-Boltzmann constant, I the electric current through the wire, and F its resistance per unit length whose temperature dependence is given by the equation F(T) ) F0(1 + a∆T + b∆T2), where ∆T ) T - 300 (K) and F0 is the resistance at 300 K.20 The values of the transport coefficients and other constants are given in Table 1. The radiation term ∼σT4 is essentially negligible at the present low temperatures but included for completeness. For the mass transport, we can distinguish qualitatively different cases: Since there is a net axial flow through the reactor with a fairly low velocity, the mass transport is a combination of this axial flow (dominating far away from the catalyst wire) and diffusion (dominating close to the catalyst). For sufficiently large flow velocity and small conversion rate (i.e., low temperature) the diffusive transport region is essentially a cylindrical region around the catalyst with radius Rn ∼ Di/ Vg,12 where Di is the diffusion coefficient for reactant i and Vg the linear flow velocity. In the present case, the situation prior to and exactly at ignition corresponds to this simple cylindrically symmetric case, with Rn smaller than the reactor tube radius, since the reaction rate is small and the gradients are weak or nearly nonexistent. When, in contrast, the reaction rate is high, significant gradients may extend up to the tube radius, and the diffusive transport then becomes important over large distances

and asymmetric with respect to the up- and downstream sides of the catalyst. In our calculation we have for simplicity assumed the cylindrical symmetry case with Rn ) 0.5 cm (i.e., approximately half the tube radius), which is a good representation of the preignition and early postignition regimes, where we thus expect the mass transport to be described quite well. Comparing experiments with calculation, we should, however, be aware that this model may oVerestimate the efficacy of the mass transport, especially in the late postignition regime, since the mass transport tends to be successively more one-dimensional if the reaction rate is high and the gradients extend axially to larger distances than the tube radius. To account for this deviation from cylindrical symmetry, we introduce a geometry correction factor, γ (see below). There is also a counteracting factor to the latter mass transport restriction, namely convection which is not included in the model: When the reaction rate is high, the temperature of the catalyst becomes high and the temperature gradient near it becomes strong. This will create convective flow, enhancing mass transport around the catalyst, not accounted by our chosen cylindrically symmetric diffusion flux. The latter effect is small near the ignition point when the difference between the catalyst and inlet gas temperatures is small but of increasing importance at later stages. In summary, we choose a model for mass transport judged to be well representative near the ignition point, being aware that it may be successively poorer in the late postignition regime, where it overestimates diffusive mass transport. In addition, we should define the radii of the above-mentioned cylindrical regions and the boundary conditions at their outward surfaces. These radii obviously cannot be larger than the radius of the reactor tube. On the other hand, if the average velocity of the gas in the tube, Vg, is high, the regions where the mass transport is dominated by diffusion are confined close to the wire, and their extensions can be estimated as Di/Vg from the scaling arguments.12 For our experimental conditions, Vs < 2 cm s-1, which yields values of the radii of the cylindrical regions in the range 0.5-1 cm. We use the lower of these values, Rn ) 0.5 cm, to be compared with the radius of the tube which is 0.95 cm (for the sake of simplicity, we assume the radii for the two reactants to be equal). This value of Rn also agrees well with space-resolved measurements of the concentration profiles of the reactant gases for a slightly different experimental geometry.18 The boundary conditions for the concentrations can then be obtained by noting that the inward diffusive fluxes inside the cylinder, which is 2πRnDi∂ni(Rn)/∂r per unit length, are equal to the differences in the amounts of the reactants borne by the incoming and outgoing flows, 2RnγVg[n0i - ni(Rn)], where γ is the above-mentioned geometry correction factor of order unity and n0i are the concentrations in the incoming mixture. (The factor γ is introduced to take into account deviations from cylindrical symmetry in the gas concentrations.) These considerations yield the following equation:

π Di

∂ni(Rn) ) γVg[n0i - ni(Rn)] ∂r

(3.12)

The boundary conditions (eq 3.12) are expected to approximately describe the mass transfer both before and after ignition. They are always valid at the initial stages of ignition, when the gradients in the reactant concentrations are small and the concentrations near the wire are close to those in the incoming flow. If Vg is large enough to localize the diffusiondominated region around the wire even at the high reaction rates



established after ignition, eq 3.12 still provides a satisfactory approximation. If, however, Vg is small or zero, depletion of the reactants may extend up to the tube walls and lead to a steady state where the mass transport is rather limited by diffusion along the axis of the tube and hence the described model is not reflecting the experimental situation anymore. The situation with the heat transport is simpler because the heat conduction is always dominated by the heat transfer to and through the walls of the tube maintained at the room temperature. Therefore the boundary condition for the temperature is taken to be

T(RT) ) T0

(3.13)

where T0 is the room temperature and RT ) 0.95 cm is the radius of the tube. This value of RT is also in accordance with the measured temperature profiles.18 The validity of this approximation is subject to similar considerations as for the mass transport. At and near the ignition point the validity is best, and the approximation is not critical since the T-gradients are then weak anyway. C. Calculation of Ignition Temperature and Ignition Traces. The ignition temperatures for different gas mixture compositions have been calculated by the procedure described in ref 17. In short, for given electric current and wire temperature, we calculate the steady-state heat loss function Ql (from the temperature gradient) and the concentrations of the reactants near the wire (taking into account the gradients in the concentrations), and we calculate the reaction rate and the corresponding chemical power release Qc via the kinetic model.17 The possible steady states of the system correspond to the temperatures at which Ql ) Qc. As the electric current (temperature) is increased, one may reach a critical current and temperature, Iign and Tign, respectively, where no steady-state solution exist. This is the point where the ignition occurs.1,12 To calculate an ignition trace (i.e., the temporal self-evolution after ignition), we solve the system of the time-dependent equations (eqs 3.7-3.13) with a slightly supercritical value of the electric current I ) Iign + δI, where δI , Iign, and with the initial stationary profiles of temperature and concentrations corresponding to a slightly subcritical current I ) Iign - δI. D. Numerical Procedure. The boundary conditions (eqs 3.9-3.11) are nonlinear with respect to ni and T, mainly because the dependence of the reaction rate W on the reactant partial pressures and temperature is highly nonlinear. Besides that, the concentrations of the reactants undergo very rapid spatiotemporal changes in the course of ignition. Under such circumstances, the standard Crank-Nicholson method21 for numerical solution of transport equations (eqs 3.7 and 3.8) is not directly applicable. For this reason, we have used the following numerical procedure: Each time step t f t + ∆t is first made by the CrankNicholson method with the fixed reaction rate W(t) found at the previous step. The new values ni(t + ∆t) and T(t + ∆t) obtained in this way are used to find W(t + ∆t) from eqs 3.13.6. After that, the step t f t + ∆t is repeated with this new value of W. The iterations are proceeded until a convergence of the reaction rate at each time step is reached. Such an approach guarantees the stability of the numerical scheme at all the stages of ignition. The method of numerical solution of the kinetic equations (eqs 3.1-3.6) is described in ref 12. In the case of existence of three solutions for a given gas mixture, we do not consider the unstable one corresponding to the intermediate value of the

Figure 1. Ignition temperature as a function of the relative concentration of H2 in the gas mixture with PH2 + PO2 ) 45 Torr, total pressure 760 Torr, and nitrogen as a carrier gas. The dots and the solid line show the measured and calculated values, respectively. The experimental data are the same as in ref 12.

reaction rate. When a stable kinetic steady state, that we follow, disappears (i.e., the system undergoes a kinetic phase transition), we proceed to the other stable branch of the solutions of eqs 3.1-3.6. We find the time step ∆t ) 10-3 s sufficient for convergence of the numerical solution. To reduce eqs 3.7 and 3.8 to onedimensional form, we use x ) log(r/r0) as an independent variable with step ∆x < 10-2. Since it is hard to evaluate the geometry correction factor γ a priori, we fit its value to best reproduce the ignition trace at one particular experimental condition, R ) 0.5, Φ ) 150 cm3 min-1, and nitrogen as a carrier gas. This gives γ ) 2/3 which is fairly close to 1. This is the only fitting parameter introduced, and we use this single value of γ, derived for the single gas mixture, in all simulations at all other conditions. 4. Results and Discussion We first present the ignition temperatures, then the postignition kinetics, and finally discuss the role of kinetic phase transitions and the mechanisms of self-extinction and selfsustained oscillations within the present model. A. Ignition Temperature. The calculated ignition temperature, Tign, versus relative concentration of hydrogen in the gas mixture, R, for N2 as a carrier is shown in Figure 1 by the filled line, together with the experimental points (filled circles). The calculated values of Tign are slightly different (lower for lower R) from those reported in ref 12 because of a slight difference in the transport coefficients and boundary conditions used. The experimental points are the same as in ref 12. The figure demonstrates a good agreement of the calculated Tign with the observed ones. The sharp decrease of the ignition temperature with decreasing R is due to the formation of gradients in the hydrogen concentration, which makes the local value of R near the wire smaller than that in the original mixture and facilitates ignition.12,17 At R ≈ 0.02, the theoretical curve reaches 300 K; i.e., the reaction ignites already at room temperature. In the experiment, such a behavior was also observed, but for slightly higher hydrogen content 0 < R j 0.1. At high R-values the ignition temperature increases due to a combination of hydrogen poisoning and gradients in O2, which in turn results in a local R-value at the catalyst higher than in the feed gas.

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Figure 2. Measured (left panels) and calculated (right panels) ignition traces for different relative concentrations of hydrogen in the gas mixture, R (indicated on the right panels). The data correspond to zero net flux of the gas mixture through the tube (Φ ) 0) with PH2 + PO2 ) 45 Torr, total pressure 760 Torr, and nitrogen as a carrier gas. For R ) 0.2 (the upper left panel), experimental data are not available.

B. Ignition Traces. The measured and calculated ignition traces for N2 and He as carriers, at different values of the net flux through the tube, Φ, and different relative hydrogen concentrations, R, are presented in Figures 2-5. These figures demonstrate that the calculations quite well reproduce the ignition temperatures, the asymptotic temperatures in the mass transport-limited regime, and, in most cases, the temperature overshoot. More detailed analysis reveals the following general trends in the ignition process: With increasing R (in all the figures from upper to lower panels), the initial (ignition) temperature increases in agreement with Figure 1 (note that Tign is not particularly sensitive to the net flow). The asymptotic (postignition) steady-state temperature demonstrates a nonmonotonic dependence on R. At low values of R, ignition leads to a depletion of hydrogen in the vicinity of the wire, and the reaction comes to a regime where its rate is totally controlled by the diffusive transport (of H2). At high values of R, oxygen becomes the species depleted by reaction, and the final steady state is controlled by the diffusion of oxygen. Accordingly, the asymptotic temperature decreases with increasing R at large R. At some intermediate value, R ) R*, the final temperature reaches its maximum. If we neglect the contribution of the electric heating to the final temperature, this R* can be estimated (cf. the analysis in ref 12) from the condition

nH2(Rn) ≈ nO2(Rn) ≈ 0

(4.1)

which with the help of the corresponding stationary solutions of eqs 3.7,

Figure 3. Same as in Figure 2 with Φ ) 150 cm3 min-1.

ni(r) ≈ ni(Rn)

log(r/r0) log(Rn/r0)

ni(Rn) ∂ni(r) ≈ (4.2) ∂r r log(Rn/r0)

and the boundary conditions (3.9, 3.10, 3.12) yields

R* )

2DO2 + 2η DH2 +2DO2 + 3η

(4.3)

with η ) πDH2DO2/[γVgRn log(Rn/r0)]. For large η (zero flow limit), the diffusion limitations become irrelevant, and the total chemical power release is determined by the mixture composition. Equation 4.3 gives then the stoichiometric value R* ) 2/3 (as it would do if DH2 were equal to DO2). For the case of large flow, the limiting case of eq 4.3, R* ) 2DO2/(DH2 + 2DO2) (0.34 and 0.49 for nitrogen and helium as carrier gas, respectively), is easy to obtain directly from (4.2) by assuming ni(Rn) ≈ n0i . For nitrogen as carrier gas and Φ ) 0, 150, and 300 cm3 min-1, eq 4.3 yields R* ) 0.67, 0.57, and 0.52, respectively. For helium as carrier gas and Φ ) 150 cm3 min-1, eq 4.3 predicts R* ) 0.63. These values agree with experiment (compare the final temperatures for different R values at the left panels of Figures 2-5). All deviations of R* from the stoichiometric value 2/3 are due to the fact that DH2 > DO2. Another prominent feature that can be seen in both the experimental and the theoretical sequences in Figures 2-5 is a nonmonotonic dependence of the width of the temperature peak on R. The peak first becomes sharper with increasing R, while for R ) 0.67 it broadens again; i.e., the transition to the postignition state is extended in time and the T versus t curve becomes much smoother (Figures 2-4). This can be explained by a kinetic phase transition that follows ignition for smaller values and is absent for larger values of R. (The relationship between ignition and kinetic phase transitions is discussed in detail in the following subsection.)



Figure 5. Same as in Figure 2 with Φ ) 150 cm3 min-1 and helium as a carrier gas. Figure 4. Same as in Figure 2 with Φ ) 300 cm3 min-1.

Since the reaction rate in the postignition state is controlled by the transport of one of the reactants, the final (asymptotic) temperature must depend on the net flux Φ. For Φ ) 0, the asymptotic postignition temperature is equal to the initial temperature, Tign, due to the total depletion of reactants in the tube (neglecting that H2 and O2 have been converted to H2O). As the reaction rate (and hence the chemical power release) in the postignition regime is determined by the reactant transport to the wire, the final temperature must always increase with increasing Φ. This tendency is indeed seen in Figures 2-4 for all values of R presented. The agreement between theory and experiment is very good for larger values of Φ and nitrogen as a carrier gas (Figures 3 and 4). For Φ ) 0 (Figure 2), the calculation underestimates the width of the temperature peak. This is because our boundary condition, eq 3.12, assumes that, at Vg ) 0, the hydrogen available for reaction is contained in the cylinder of the radius RH. In reality, the diffusion and gradient pattern changes after depletion of reactants in this cylindrical domain. At later stages the reactants are transported from the further away parts of the tube, eventually in a more or less one-dimensional manner. The width of the temperature peak is then determined by the length of the tube. This is not accounted for by the model and causes discrepancies between the measured and calculated peak widths and asymptotic temperatures at the lower flow rates. For He as the carrier gas and lower values of Φ (Figure 5), the large diffusion coefficient of hydrogen in helium makes our assumption about the approximate cylindrical symmetry of the concentration fields even less justified. This leads to an underestimation of the postignition temperature in the calculation. However, the qualitative features in the calculated and measured ignition traces in Figure 5 are still strikingly similar. In general, the calculated ignition traces give a good overall account for the measured T versus t traces, and the observed

deviations can be mainly attributed to geometric factorssnot included in the calculationsswhich influence the mass transport. C. Ignition and Kinetic Phase Transitions. The question about the relationship between ignition and kinetic phase transitions in heterogeneous catalytic reactions is important but complex. Already in ref 3 it was suggested (on the basis of the analysis in ref 1) that ignition is the result of thermal instability in the system and hence not necessarily involves bistable kinetics. On the other hand, Harold and Garske,8,9 while paying great attention to the heat transfer along the wire, ignored the thermal instability issues. Their analysis seems to imply that ignition and extinction are results of kinetic phase transitions on the catalyst surface. It has also been noted5 that neglection of a correct description of the mass transport may falsify the bistability diagrams. In another calculation, kinetic phase transitions were seen both before and after the maximum of the catalyst temperature or were not seen at all22 depending on the details of the input parameters. In principle, a sudden kinetic phase transition in the preignition regime could change the reaction kinetics from low turnover to high turnover, which in turn could cause ignition. (In the present case this would happen in a kinetic phase transition from a hydrogen to an oxygen covered surface, since hydrogen is poisoning the reaction.) For the present system this would be possiblesif at allsnear the kinetic phase transition regime for the H2/O2/Pt system, which is around R ∼ 0.10.2.12 Experimentally this would be best explored by successively decreasing R with a starting temperature just below the ignition temperature (see Figure 1). Kinetic phase transitions may more easily occur in the postignition regime, but there the effect will not be as dramatic as if ignition is caused by a kinetic phase transition just at ignition (except if it causes extinction). The numerical simulations make it possible to shed light on the relationship between ignition and kinetic phase transitions because in the calculation it is possible to track many physical

3176 J. Phys. Chem. B, Vol. 103, No. 16, 1999 values at the same time, which is more difficult in the experiments. Maybe the most important quantity to follow is the surface coverage of oxygen and hydrogen, since a kinetic phase transition would be connected with a sudden change in both these coverages.2 In this section we discuss two different scenarios of ignition and give numerical examples for one of them. D. Ignition Scenarios. We consider the case when we are in the preignition regime and in the so-called low-reactive state, with a predominantly hydrogen-covered surface,2 but quite close to the kinetic phase transition point at R ∼ 0.1-0.2. (At lower R the system starts in the high-reactive state (oxygen-covered surface) already in the preignition regime, and ignition can therefore not be caused by a kinetic phase transition there. At much higher R, the point of the kinetic phase transition cannot be reached since the surface is always predominantly covered by hydrogen.) As the temperature is raised by a small stepwise current increase, different possible scenarios can be distinguished: (i) The increase in the heat loss due to the larger T gradient compensates fully the increase in the chemical power release. After a transient adjustment of the gradients the system reaches a new steady state and no ignition takes place. (ii) The increase of the chemical power release is not fully compensated by the increase of the heat loss function. In this case, all the reaction heat cannot be dissipated anymore in a new preignition steady state, and the temperature starts to selfaccelerate. This ignition situation still occurs on the low-reactive branch of the reaction (hydrogen-dominated surface). Thus there is no kinetic phase transition at ignition. The gradual further increase of the temperature (by self-heating) may (or may not) change the hydrogen adsorption-desorption equilibrium enough for a kinetic phase transition to an oxygen-covered surface to occur. However, since ignition occurs prior to the kinetic phase transition (if at all occurring), the latter is not causing the ignition event. (iii) Finally, the increase of the heat loss function may be sufficient to neutralize the increase of the chemical power release only as long as the system stays on the low-reactive branch. However, the adsorption-desorption equilibrium for hydrogen may be disturbed due to the change in the local composition of the gas mixture (becoming more oxygen-rich, since we are in the H2-deficit regime where H2 gradients develop more strongly than O2 gradients). This will cause a change toward smaller values in the local R-value at the catalyst, which in turn can cause a kinetic phase transition from a hydrogen-covered to an oxygen-covered surface before ignition has occurred. In this case, the reaction rate will increase rapidly since it jumps to the high-reactive state and so will the temperature of the wire. In this case ignition is induced by a kinetic phase transition. Since the reaction rate almost always increases exponentially with temperature (Arrhenius temperature dependence), while the change in the hydrogen concentration with varying R is linear, the scenario ii must be much more abundant. However, the scenario iii may in principle occur when the system initially is very close (in R) to the regime of bistable kinetics. In our case, the bistable region stretches to very low values of the relative hydrogen concentration (a few percent). At these concentrations, the dependence of the reaction rate on temperature is so steep that ignition occurs immediately after letting the reactant mixture into the reactor even without electric heating (in Figure 1, to the left from the intersection of the curve with the R axis). This means that for given temperature and pressure of the reactants, the ignition always occurs via scenario ii.

Kulginov et al.

Figure 6. Time dependence of the temperature (upper panel), reactant concentrations reaction near the surface of the wire (middle panel), and reaction rate (lower panel, 1 ML ) 2.5 × 1015 molecules/cm2) during ignition. The conditions correspond to the relative concentration of hydrogen R ) 0.2, the net flux of the mixture Φ ) 150 cm3 min-1, and nitrogen as a carrier gas. The two vertical dashed lines show the ignition point (the inflection point of the temperature, left) and the kinetic phase transition point (right).

This is illustrated by Figure 6 that shows the calculated time dependences of the temperature, reaction rate, and the hydrogen and oxygen coverages for two different relative concentrations of hydrogen, R, in the reactant mixture. Both calculations correspond to the mixture net flux Φ ) 150 cm3 min-1. The points of ignition (identified as the inflection points for the temperature) and kinetic phase transition are marked by vertical dashed lines. For R ) 0.2 (Figure 6), the ignition occurs well before the kinetic phase transition. The latter corresponds to the sharp maximum of the reaction rate. The rapid self-acceleration of the reaction after ignition causes almost complete depletion of hydrogen near the surface of the wire (Figure 6, the middle panel), which in turn induces the kinetic phase transition to an oxygen-covered surface, but since there is no hydrogen available, the reaction comes to a regime limited by hydrogen diffusive transport and the reaction rate sharply decreases again (Figure 6, the lower panel). However, the system is still in the high-reactive state and the oxygen coverage dominates the surface. Note that the maximum of the temperature (Figure 6, the upper panel) is delayed with respect to the maximum of the reaction rate. This delay is caused by an inertia in the formation of the temperature gradient. Right after the kinetic phase transition, the temperature gradient is still weak and the reaction mainly heats the wire, whose temperature therefore continues to grow. After about 1 s, the formation of the temperature gradient is essentially completed and the heat is efficiently


Figure 7. Same as in Figure 6, but for the relative concentration of hydrogen R ) 0.6. The vertical dashed line shows the ignition point.

transported away from the wire. This causes the decrease of the temperature. For R ) 0.6 (Figure 7), the species depleted by the reaction is oxygen (the middle panel). Accordingly, the relative concentration of hydrogen increases after ignition, and the surface remains covered by hydrogen. For that reason, the growth of the reaction rate after ignition is not so rapid (the lower panel), and oxygen is not completely depleted near the wire. This makes the transition from the kinetic- to the diffusion-controlled state much more gradual, which is manifested in a smooth and broad maximum in the temperature dependence (the upper panel). The system finally stabilizes in the low-reactive state with domination of hydrogen coverage. At large R and small values of Φ, the system can finally undergo a kinetic phase transition to an oxygen-covered surface well after the temperature maximum, because hydrogen at such conditions is depleted more rapidly than oxygen (as discussed in ref 22). E. Extinction and Oscillatory Behavior. For certain experimental conditions, a spontaneous extinction of once ignited catalytic oxidation has been reported (see, e.g., ref 18). Sometimes, after such an extinction, the system comes to ignition again, which gives rise to more or less regular oscillations of the temperature of the wire which can be described as repeated ignition-extinction cycles. We demonstrate here that, for some conditions, spontaneous ignitionextinction cycles are possible to find in numerical simulations. Figure 8 shows the calculated time dependence of the temperature, of the reactant concentrations near the wire, and of the reaction rate for R ) 0.2, Φ ) 0, and helium as a carrier gas. The external radii for the reactant concentrations in the calculation are Rn ) 2 and 0.5 cm for hydrogen and oxygen, respectively. First (the left-hand part of the Figure 8) a typical


Figure 8. An ignition-extinction sequence. The temperature (upper panel), the concentrations of H2 and O2 near the wire (middle panel), and the reaction rate (lower panel) are shown. The two vertical dashed lines show the points of kinetic phase transitions from hydrogen- to oxygen-covered surface (left line) and back (right line), respectively. The conditions correspond to the relative concentration of hydrogen R ) 0.2, net flux of the mixture Φ ) 0, and He as a carrier gas. The external radii for the reactant concentrations used in the calculation are RH ) 2 cm and RO ) 0.5 cm.

ignition-kinetic phase transition sequence occurs. Because of the reaction stoichiometry and small R, the rapid increase of the reaction rate in the course of ignition brings the concentration of hydrogen almost to zero while there is still plenty of oxygen in the vicinity of the wire. At this point (indicated by the left vertical dashed line), the kinetic phase transition from hydrogento oxygen-covered surface occurs. After that, the reaction rate is controlled by the diffusion of hydrogen, as discussed in the previous subsection. However, the large diffusion coefficient for hydrogen (twice as large as that for oxygen) allows comparatively high reaction rate in this diffusion-limited regime. In this particular situation, the diffusive transport of oxygen to the surface of catalyst cannot follow its consumption by the reaction, which leads to a gradual depletion of oxygen in the vicinity of the wire (the middle part of the Figure 8). As can be seen in the figure, the concentration of oxygen near the surface of the catalyst decreases while the hydrogen concentration stays almost constant. Accordingly, the relatiVe concentration of hydrogen increases up to the point where adsorption of hydrogen “wins” over adsorption of oxygen and a reVerse kinetic phase transition to a hydrogen-dominated surface occurs (indicated by the right vertical dashed line). As a consequence, the reaction rate and the temperature sharply decrease (spontaneous extinction of the reaction). In contrast to the smooth temperature growth after the first kinetic phase transition (the left part of the upper panel of Figure 8), the temperature decreases abruptly after the extinction of the

3178 J. Phys. Chem. B, Vol. 103, No. 16, 1999 reaction. This can be explained by the considerable temperature gradient at the moment of extinction. Due to the difference in Ri (RH > RO), there is more hydrogen than oxygen left after extinction, and due to the larger diffusion coefficient for hydrogen, its concentration at the catalyst surface increases more rapidly than that of oxygen. Therefore, the relative concentration of hydrogen at the surface continues to increase and ignition never happens again (the right-hand part of the Figure 8). The key point behind the discussed behavior is the depletion of oxygen after the kinetic phase transition. If either the total amount of hydrogen is not sufficient to sustain the necessary reaction rate long enough (smaller values of Rn for hydrogen) or the oxygen supply is sufficient to prevent its depletion (larger values of Φ), the spontaneous extinction does not occur in the calculation. It is thus the result of a subtle interplay between reactants transport to the reaction zone and kinetics. Although interesting as a complex behavior caused by a combination of bistable kinetics with diffusion limitations (cf. refs 23 and 24), it lacks the generality that is necessary to account for the wealth of oscillatory behavior observed.18 Most probably, the ignitionextinction oscillations are explained by changes in the catalyst properties, e.g., restructuring of its surface or oxide formation25 in the course of reaction, or some other modification/contamination. 5. Conclusions We have performed experimental studies and numerical simulations of ignition of catalytic oxidation of hydrogen at the surface of a platinum wire. Our main aim was to study the factors that govern the transient behaVior and the postignition steady state. In the calculations, we used a kinetic reaction model based on the existing surface-science data and a linear model of heat and mass transport. A good agreement between observed and numerically simulated transient behavior of the catalyst temperature was found. The experimental postignition temperatures are reproduced in the calculations with only one fitting parameter in the model (a geometry correction factor). We have also studied and discussed the role of kinetic phase transitions on ignition. We discussed three possible scenarios of catalytic ignition involving kinetic phase transitions and numerically illustrated two of them. Furthermore, spontaneous

Kulginov et al. extinction of the ignited reaction, caused by diffusion limitations of hydrogen, was discussed and was found to be quite exotic. As a general conclusion, catalytic ignition has come to a state of thorough understanding by a combination of experiments and detailed surface kinetics models and numerical simulations. Acknowledgment. We gratefully acknowledge many useful comments by Prof. V. P. Zhdanov (Boreskov Institute of Catalysis, Novosibirsk, Russia) on the manuscript. This work has been done with financial support of NUTEK, Grant Dnr. 95-11665. References and Notes (1) Frank-Kamenetskii, D. A. Diffusion and Heat Transfer in Chemical Kinetics, 2nd ed.; Plenum: New York, 1969. (2) Zhdanov, V. P.; Kasemo, B. Surf. Sci. Rep. 1994, 20, 111. (3) Schwartz, A.; Holbrook, L. L.; Wise, H. J. Catal. 1971, 21, 199. (4) Schefer, R. W. Comb. Flame 1982, 45, 171. (5) Sheintuch, M.; Schmidt, J. Chem. Eng. Commun. 1986, 44, 33. (6) Cho, P.; Law, C. K. Comb. Flame 1986, 66, 159. (7) Kaul, D. J.; Sant, R.; Wolf, E. E. Chem. Eng. Commun. 1987, 42, 1399. (8) Harold, M. P.; Garske, M. E. J. Catal. 1991, 127, 553. (9) Garske, M. E.; Harold, M. P. Chem. Eng. Sci. 1992, 47, 623. (10) Ikeda, H.; Libby, P. A.; Williams, F. A.; Sato, J. Comb. Flame 1993, 93, 138. (11) Vlachos, D. G.; Schmidt, L. D.; Aris, R. Comb. Flame 1993, 95, 313. (12) Fassihi, M.; Zhdanov, V. P.; Rinnemo, M.; Keck, K.-E.; Kasemo, B. J. Catal. 1993, 141, 438. (13) Rinnemo, M.; Fassihi, M.; Kasemo, B. Chem. Phys. Lett. 1993, 211, 60. (14) Warnatz, J.; Allendorf, M. D.; Kee, R. J.; Coltrin, M. E. Comb. Flame 1994, 96, 393. (15) Ikeda, H.; Sato, J.; Williams, F. A. Surf. Sci. 1995, 326, 11. (16) Vlachos, D. G.; Bui, P.-A. Surf. Sci. 1996, 364, L625. (17) Rinnemo, M.; Kulginov, D.; Johansson, S.; Wong, K. L.; Zhdanov, V. P.; Kasemo, B. Surf. Sci. 1997, 376, 297. (18) Lundgren, S.; Keck, K.-E.; Kasemo, B. ReV. Sci. Instrum. 1994, 65, 2696. (19) Hellsing, B.; Kasemo, B.; Zhdanov, V. P. J. Catal. 1991, 132, 210. (20) Landolt, H. Constants and Functions in Science and Engineering; Springer: Berlin/Heidelberg, 1982. (21) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C; University Press: Cambridge, U.K., 1994. (22) Rinnemo, M.; Deutschmann, O.; Behrendt, F.; Kasemo, B. Comb. Flame 1997, 111, 312. (23) Zhdanov, V. P.; Kulginov, D.; Kasemo, B. Phys. ReV. 1996, E53, R3013. (24) Kulginov, D.; Zhdanov, V. P.; Kasemo, B. J. Chem. Phys. 1997, 106, 3117. (25) Zhdanov, V. P. Surf. Sci. 1993, 296, 261.

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