Temperature Fronts and Patterns in Catalytic ... - ACS Publications

Ind. Eng. Chem. Res. 1997, 36, 2931-2944

2931

Temperature Fronts and Patterns in Catalytic Systems Dan Luss Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792

Current understanding and open questions concerning formation and motion of temperature fronts and patterns in packed-bed reactors and on single catalytic pellets are reviewed. For the single-reaction case, it is possible to readily predict the maximal temperature of fronts formed in packed-bed reactors following sudden changes in the feed conditions (wrong-way behavior) or during reverse-flow operation. There is a need to extend these relations for systems involving multireactions and reversible changes in the pellet state. Complex temperature patterns may be generated on catalytic surfaces by global interaction and surface nonuniformities. Simulations predict that even more intricate patterns may evolve in packed-bed reactors. It is still unresolved which of these complex patterns exist in commercial reactors, what is their impact, and which control policies can stabilize beneficial temperature patterns. The predictions of several important dynamic features, such as hot-spot formation, await further analysis of models which account also for the momentum transport and changes in the physical properties of the fluid in the reactor. Introduction Catalytic packed-bed reactors have served as the workhorse of the chemical and petrochemical industries for many decades. The interest in the dynamics of catalytic pellets and packed-bed reactors has been recently rejuvenated by the development of novel reactor configurations and modes of operation, new mathematical tools, enhanced computational capabilities, and new experimental techniques. The need to enhance our understanding of the dynamics of packed-bed reactors has received additional emphasis with the application of this reactor technology as a primary tool for pollutant destruction, as these reactors need to maintain a high level of conversion even when subjected to rapid and large variations in the flow rate and feed composition. Gilbert Froment has made many pioneering contributions to the design and analysis of packed-bed reactors and to the dissemination of that knowledge. He has organized and chaired many professional meetings which stimulated novel advances and applications. He has just organized and chaired a most stimulating and interesting ISCRE meeting and is already involved in organizing a new one on “Dynamics of surfaces and reaction kinetics in heterogeneous catalysis.” To honor Prof. Froment’s birthday and his many contributions, I would like to review briefly some problems associated with the evolution, propagation, and impact of temperature fronts in heterogeneous catalytic systems. The topics covered will be wrong-way behavior, temperature rise in reverse-flow reactors, spatiotemporal temperature patterns on single catalytic pellets and in packedbed reactors, thermoflow multiplicity, and hot-spot formation. While our understanding of these problems has advanced significantly, there are still many important and interesting open questions. Due to space limitations, this review focuses on the current understanding rather than covering all the relevant literature. Thus, I apologize in advance to the many researchers whose important contributions are not covered here. Wrong-Way Behavior The wrong-way behavior is a unique and counterintuitive response of a packed-bed reactor to a rapid change in the feed conditions. For example, following S0888-5885(96)00597-0 CCC: $14.00

Figure 1. Wrong-way behavior in an adiabatic fixed-bed methanator following a step decrease in the feed temperature from 231 to 214 °C. After Van Doesburg and DeJong (1976).

a sudden decrease in the feed temperature, the reactant concentration increases in the upstream section of the reactor. This generates, in turn, a transient temperature-rise in the downstream section of the reactor. This response is caused by the difference in the propagation speed of the concentration and temperature disturbances in the reactor. This surprising temperature-rise may damage the catalyst and/or the reactor and complicate the design of a proper control. Van Doesburg and DeJong (1976) observed such a response during the methanation of a mixture of carbon monoxide and carbon dioxide in an adiabatic reactor (Figure 1). Under conditions for which multiple steady states exist, the transient temperature-rise may ignite the reactor and shift it permanently to a high-temperature, highconversion steady state, as observed by Sharma and Hughes (1979) during the oxidation of carbon monoxide (Figure 2). In the case of several simultaneous reactions, the wrong-way behavior may initiate undesired, highly exothermic reactions, which may lead to a disastrous runaway or even an explosion. Matros and Beskov (1965), Boreskov and Slinko (1965), and Crider and Foss (1966) were the first to discover this response. The early experimental obser© 1997 American Chemical Society

2932 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997

In general, the dimensionless velocity of the thermal wave

w)

zLe tu

Le )

(Fcp)g + (1 - )(Fcp)s

(4)

(Fcp)g

is about unity. Chen and Luss (1989) have shown that when w deviates significantly from unity, the approximation of λe is improved by multiplying the second term on the right-hand side of eq 1 by w. The approximation is not expected to be accurate when w is much smaller than unity, and it fails when w is negative, i.e., for a backward (upstream) movement of the wave. In such cases, the reactor dynamics should be simulated by a two-phase model that accounts for the thermal dispersion. When a unique steady state exists for all feed temperatures, the impact of the wrong-way behavior is important only if the initial conversion is either intermediate or high and the temperature front is at some intermediate location in the reactor (Pinjala et al., 1988; Chen and Luss, 1989). In all the other cases the transient temperature increase is negligible. Simulations show that following a large step decrease in the feed temperature the transient peak temperature approaches an asymptotic value of T*. This value depends on the dimensionless speed of propagation of the wave, namely

T* - Tf ) Figure 2. Ignition of an extinguished state by the wrong-way behavior following a sudden decrease in the feed temperature to an adiabatic reactor in which carbon monoxide is oxidized. After Sharma and Hughes (1979).

vations of this behavior include those of Hoiberg et al. (1971), Van Doesburg and De Jong (1976), Sharma and Hughes (1979a,b), and Oh and Cavendish (1982). Several predictions of the magnitude of the wrongway behavior exist for the single-reaction case. The simplest predictions are obtained using a pseudohomogeneous plug-flow model (Mehta et al., 1981). Unfortunately, this model often predicts unrealistically hightemperature excursions. Proper predictions are usually obtained by a pseudohomogeneous model, using an effective thermal dispersion to account for the impact of both the thermal conductivity of the solid phase, λs, and the heat-transfer resistance between the two phases. Vortmeyer and Scha¨fer (1974) predicted that a proper overall effective thermal dispersion should satisfy the relation

λe ) (1 - )λs + λsg

(1)

where

λsg )

u2(Fcp)2g hav

(2)

A two-phase model that ignores the axial thermal dispersion will predict a similar behavior if one uses a modified heat-transfer coefficient h*, satisfying the relation

λeav 1 ) h* u2(Fc )2 p g

(3)

(-∆H)Cf (1 - w/Le) Ffcf 1-w

(5)

In general, Le is much larger than unity, and w/Le is small in comparison to unity. Thus, the temperature rise is associated with the slowing of the thermal front velocity by the chemical reaction. For an nth-order reaction in an adiabatic reactor, T* may be computed by the following approximate relation (Pinjala et al., 1988):

(-∆H)C2-n u2 f ) λekˆ

E ∫0RT*/Eexp(- RT )(1 - T*T )

n-1

dT (6)

For a first-order reaction, eq 6 predicts that

(-∆H)Cfu2 E E E ) T* exp - E1 λekˆ RT* R RT*

(

)

( )

(7)

where

E1(x) )

∫x∞

exp(-t) dt t

(8)

Figure 3 shows that the predicted T* values agree very well with the numerical simulations. Kiselev and Matros (1980) predicted that the peak temperature will be bounded by a linear function of log[(-∆H)Cfu2/λekˆ )]. The nonlinear dependence of T* on this term, shown in Figure 3, indicates that the proposed bound is not precise. The wrong-way behavior leads to several novel features when different steady states may exist under the same operating conditions. Again, the impact of the wrong-way behavior is negligible when the conversion is either very low or very high with a reaction front very close to the reactor inlet. The most interesting dynamic features occur when the reactor is initially extinguished

Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 2933

Figure 3. Comparison of the computed maximal bed temperature caused by the wrong-way behavior (black dots) with the prediction of eq 7 as a function of (-∆H)Cfu2/λekˆ . After Pinjala et al. (1988).

Figure 5. Comparison of the transient gas and catalyst temperatures during a wrong-way behavior which ignites an extinguished state in the presence and absence of reactant adsorption on the inert support. After Il’in and Luss (1992).

Figure 4. Ignition of an extinguished state in an adiabatic reactor following a sudden cooling of the feed. After Pinjala et al. (1988).

but close to the ignition point. A sudden, small feedtemperature decrease leads to a rather mild temperature excursion. However, a large feed-temperature decrease may lead to an ignition, similar to the behavior observed during the CO oxidation experiment shown in Figure 2. Simulations of a similar case (Figure 4) indicate that initially a backward-moving temperaturefront develops. It ignites the upstream section, generating a downstream-moving temperature wave, which shifts the reactor to the ignited state. This ignition, if not prevented by a proper control action, may lead to disastrous consequences. A very large feed-temperature decrease generates a high temperature peak at the downstream section of the reactor, which then exits very slowly (0 < w , 1) from the reactor. Reversible changes in the state of the catalyst may affect the dynamics of the reactor without affecting the corresponding steady-state behavior. For example, the reactant and/or products may adsorb reversibly on the inert support without affecting the steady-state perfor-

mance of the reactor. Thus, this rate process is usually not accounted for in models of the reactor. However, this reactant adsorption on the support may, however, affect the dynamic behavior of the reactor (Il’in and Luss, 1992). In the common case that the wrong-way behavior generates a downstream-moving temperature front, reactant adsorption moderates and decreases the peak temperature of the moving front, and slightly increases the velocity of the moving temperature-front. However, when the reactor has several steady states and is initially extinguished but close to the ignition point, a sudden decrease in the feed temperature may generate a backward-moving temperature-front and shift the reactor permanently to the ignited state. In these cases, the release of the adsorbed reactant may significantly increase the maximal temperature of the fronts. Figure 5 illustrates this effect by comparing the response of a packed-bed reactor to the same sudden decrease in the feed temperature in the presence and absence of reactant adsorption. The comparison reveals that reactant adsorption on an inert support may increase the transient temperatures, as well as the steepness and velocity of the temperature front. Moreover, the reactant adsorption generates a transient peak temperature that exceeds that of the ignited steady state. It should be noted that very strong reactant adsorption may, however, prevent ignition in cases that would have led to ignition in its absence. The impact of the wrong-way behavior in multireaction systems is usually much more pronounced than in the single-reaction case due to the taking over by undesired reactions, the rate of which is negligible under the standard operating conditions (Il’in and Luss, 1993). Clearly, the impact of the wrong-way behavior in multireaction systems is strongly dependent on the rate of the undesired reactions. Figure 6 compares the wrong-way behavior of a reactor in which a single reaction, A f B, occurs with one in which the consecutive reactions A f B f C occur, with B being the desired product. The steady-state temperature and yield of B


Figure 6. The impact of an undesired reaction (B f C) on the transient catalyst temperature-rise following a sudden decrease in the feed temperature. After Il’in and Luss (1993).

of both the initial and final steady states are essentially unaffected by the presence of the undesired reaction B f C. Yet the simulations showed that, following a sudden decrease in the feed temperature, the undesired reaction slowed the temperature-front movement and significantly increased the temperature excursion. While for a single reaction the transient peak exceeded the adiabatic steady-state temperature rise by 46%, for the two reactions the increase was more than 150%. It should be pointed out that a wrong-way behavior may follow not only a sudden change in the feed temperature but also a sudden increase in feed flow rate (Il’in and Luss, 1992, 1993). Thus, special care to avoid this response needs to be exercised in the operation of packed-bed reactors for pollutant destruction, as they need to adjust to rapid changes in both the load and feed composition. In general, one may moderate the wrong-way behavior by making only gradual, slow changes in the feed temperature and/or velocity and avoiding a sudden one. While a rather comprehensive understanding of the impact of the wrong-way behavior in the single-reaction case exists, it is essential to enhance our knowledge of its impact on various multireaction systems and those in which reversible changes in the state of the catalyst affect the reactor dynamic response. Reverse-Flow Operation Various forms of forced periodic operation of chemical reactors have been proposed by many researchers. Comprehensive reviews were presented by Bailey (1977), Matros (1985, 1989), Stankiewicz and Kuczynski (1995), and Silveston et al. (1995). Only two conceptual types of periodic packed-bed processes have found technological applications so far. In the first, the catalyst undergoes periodic changes as it is circulated through the reactor. The first proposed application, made by Lewis et al. (1949), was a recirculating fluid-bed reactor

Figure 7. Schematic of a reverse-flow operation of a packed-bed reactor. The flow is reversed periodically by switching the two valves. After Eigenberger and Nieken (1994).

in which a catalyst was oxidized in one part of the reactor and used to selectively oxidize a reactant in another part of the reactor. Callahan et al. (1970) attempted to apply this concept to the oxidation and ammoxidation of propylene and pointed out that the economics of the process may strongly depend on the amount of oxygen that can be bound reversibly on the catalyst. If the amount is not sufficiently high, an uneconomical rate of catalyst recirculation may be required. DuPont has recently announced the application of this concept for the partial oxidation of butane to maleic anhydride (Contractor et al., 1987). Catalyst circulation occurs also in circulating bubble columns and gas-lift reactors, widely used to carry out liquid-phase hydrogenations and bioreactions. In these reactors the circulated catalyst is periodically exposed to different dissolved gas concentrations. The second type of periodic reactor is the reverse-flow operation (RFO) of packed-bed reactors, which has attracted significant interest in recent years. In this process, the direction of flow in a packed-bed reactor is periodically reversed (Figure 7) to trap a hot zone within the reactor. The cold feed is regeneratively heated up by the bed as the high-temperature zone moves downstream. Before the hot zone exits the bed, the feed flow direction is reversed. The first patent for an RFO of a packed-bed reactor was issued to Cottrell (1938) about 60 years ago. However, the recent applications and interest in RFO were motivated by its successful application to SO2 oxidation in Russia (Boreskov et al., 1979; Boreskov and Matros, 1983). An independent patent was issued to Watson (1975) for application of RFO to the reduction of SO2. The RFO offers clear advantages for certain classes of reactions. For an exothermic, equilibrium-limited reaction, the declining temperature profile in the downstream section of a reverse-flow reactor increases the conversion over that attained in an adiabatic reactor. For destruction of a dilute pollutants mixture, an RFO avoids the need to add fuel in order to stabilize the reaction zone within


the reactor, and it readily adjusts to variations in the feed load and composition. For reactions involving coke formation, an RFO offers an opportunity to conduct the desired reactions in the upstream section of the reactor, while regenerating the downstream section from coke deposited in the previous half-cycle (DeGroote et al., 1995). Other applications of this periodic operation are expected to be developed in the future. A comprehensive review of the research and applications of this reactor was recently presented by Matros and Bunimovich (1996). I shall not attempt to duplicate their review and shall instead concentrate on the predictions of the temperature of the moving hot zone. The design and operation of RFO require the ability to predict several dynamic features, such as the relation between the maximal reactor temperature and the operating conditions and all the possible stable periodic states and the corresponding regions of stability. Dynamic simulations of an RFO reactor are usually lengthy, as the reactor settles on a periodic state only after several hundred flow-direction reversals. Thus, a priori criteria predictions of the dynamic features are most useful. The computational effort may be considerably reduced by using the procedure proposed by Gupta and Bhatia (1991) by directly calculating the periodic solutions by forcing the temperature profiles at the beginning and end of a period to be mirror images of each other. Salinger and Eigenberger (1996a,b) presented an efficient computational technique for finding regions with different numbers of periodic states. Khinast and Luss (1997) described a method for determining regions of parameters with qualitatively different bifurcation diagrams of the periodic states. The temperature of an RFO reactor is usually bounded between those of two limiting cases, namely, very slow and very fast switching of the flow direction. Nieken et al. (1995) found that the maximum bed temperature was rather insensitive to variations in the switching frequency when a first-order, irreversible reaction was conducted in an adiabatic reactor with no inert section. Thus, the two limiting cases provide a good estimate of the temperature rise for intermediate switching frequencies. When the switching frequency is very high, the solid temperature remains essentially stationary due to the high heat-capacity of the solid relative to that of the gas. Matros (1989) derived a set of equations for this limiting case. Eigenberger and Nieken (1994) showed that the behavior in this case is equivalent to that of a countercurrent reactor with the feed split into two equal mass streams, flowing at the same velocity but in opposite directions. Nieken et al. (1995) have shown that for a single reaction the maximum bed temperature satisfies the relation

1 kc

∫TT+∆T max

0

ad

(

)

1 1 + kc kˆ (T)

-1

dT ) 0.5∆TadNR

(9)

λsg h kc(Fcp)g λe

(10)

where

∆Tad )

(-∆H)Cf (Fcp)g

NR )

Haynes et al. (1995) independently obtained a similar relation. Somani et al. (1997a) derived a simple explicit bound on the maximal bed temperature, namely

Tmax ) Tc + 0.5∆TadNR

(11)

where Tc is the temperature at which

kˆ (Tc) ) kc

(12)

When the switching frequency is very low, a temperature front forms and moves downstream as in the wrong-way behavior. Somani et al. (1997a) showed that the maximum temperature is bounded by the explicit relation

Tmax ) Tc + 0.5(Tc - T0)

(x

1+

)

4∆TadNR -1 Tc - T0

(13)

Equation 13 is an upper-bound on the maximum temperature, and the quality of the estimate improves with increasing values of E(Tmax - Tc)/RTc2. For systems with a low adiabatic temperature-rise, the maximum temperature is close to the maximum temperature caused by the wrong-way behavior. That maximum temperature, Tmax, can be computed by the estimate of Pinjala et al. (1988) as modified by Nieken et al. (1995) to account for the external mass-transfer resistance. It satisfies the relation

1 kc

∫TT

(

max

)

1 1 + kc kˆ (T)

-1

dT ) ∆TadNR

(14)

which is rather similar in form to eq 9 for the fastswitching frequency case. When the adiabatic temperature-rise is large (order of 50 °C), the prediction of (13) is less conservative than that of (14). The explicit estimates for both the high and low switching frequencies (eqs 11 and 13) provide rather accurate predictions when

E(Tmax - Tc) RT2c

>2

(15)

Rather intricate multiplicity and dynamic features may occur when several chemical reactions occur simultaneously in a reverse-flow reactor. So far, the only case analyzed in some detail is that of two independent (Ai f Bi i ) 1, 2) parallel reactions. Ivanov et al. (1992) pointed out that three different stable periodic states may be obtained when the reactivity of the two reactions is rather different. These include one with low conversion, one with high conversion of both reactants, and one with a high conversion of one species and low conversion of the second. The first experimental observation of this multiplicity was reported by Eigenberger and Nieken (1994) for the simultaneous oxidation of propylene and propane. At the high-temperature state (Figure 8) both reactants were consumed, while at the intermediate-temperature state only the more reactive propylene was converted. A priori estimates of the maximal temperature in a reverse-flow reactor in which two simultaneous reactions occur were presented by Somani et al. (1997a). Theoretical analysis predicted and numerical simulations verified (Somani et al., 1997a) that a “doublehump” temperature profile may form when the reactivity of the two reactants are widely different and the switching frequencies are not too high. This profile evolves and exists during a part of each flow-reversal period and has a flat plateau at some intermediate temperature, say T+. The evolution of such a double


homogeneous chemically reacting systems can be explained and predicted by a simple reaction-diffusion model of the form (Turing, 1952)

ut - D∇2u ) f(u)

Figure 8. Two different stable periodic states attained in a reverse-flow monolith reactor. The feed is a mixture of propane and propylene. Feed concentrations were such that each reactant had the same adiabatic temperature-rise. After Eigenberger and Nieken (1994).

Figure 9. Formation of a “double hump” temperature profile in a reverse-flow packed-bed reactor in which two independent reactions occur, with the rate of one reaction much higher than that of the second. Both reactions have the same activation energies (15000 cal/g‚mole) and adiabatic temperature-rise (50 °C). After Somani et al. (1997a).

hump is described in Figure 9. The upper part of the temperature profile usually moves at a higher velocity than the lower temperature section, so the spatial separation between the two sections of the temperaturefront increases with time. There is a need to enhance our understanding of the behavior and potential application of an RFO of packedbed reactors, in which reversible changes in the state of the catalyst occur with a time constant of the order of the sojourn time of the front over a pellet in the bed. Moreover, knowledge of how to conduct different reactions at different sections of a reverse-flow reactor may lead to many novel applications. The simultaneous reaction and regeneration in an RFO, described by DeGroote et al. (1995), are a striking illustration of these potential applications. Pattern Formation in Catalytic Systems Extensive research effort has been dedicated to understanding the evolution and propagation of patterns in a variety of physical, chemical, and biological systems. Slin’ko and Jaeger (1994), Ertl and Imbihl (1995), and Sheintuch and Shvartsman (1996) presented comprehensive reviews of pattern formation in catalytic systems. The recent research activity in this area has been stimulated and facilitated by major advances in our ability to predict, measure, and characterize these patterns. Numerous observed patterns in living and

(16)

These models usually include simplified kinetic expressions that do not account for all the actual rate processes, but which grasp the essential features of the experimentally observed phenomena. A two-state-variable model predicts that stationary patterns may form when one variable is a slow, fast-diffusing reacting inhibitor and the second is a fast-reacting but slowdiffusing activator; i.e., the length scale of the inhibitor (Di/kˆ i)1/2 exceeds that of the activator. The prediction of the two-variable model led to conjectures that stationary temperature patterns do not form when exothermic atmospheric reactions are carried out on catalytic surfaces, as the length scale of the activator (temperature) is significantly larger than that of the inhibitor (surface diffusivity of the reactants). However, pioneering infrared imager measurements by Schmitz’s group (Pawlicki and Schmitz, 1987) revealed that stationary and spatiotemporal temperature patterns may form on single and on arrays of catalytic pellets. Since then, temperature patterns have been observed in many reacting systems, such as hydrogen oxidation (Pawlicki and Schmitz, 1987; Lane et al., 1993), ammonia oxidation (Lobban et al., 1989; Cordonier and Schmidt, 1989), methylamine decomposition (Cordonier et al., 1989), carbon monoxide oxidation (Kellow and Wolf, 1990; Qin and Wolf, 1995), and propylene oxidation (Philippou et al., 1991). Patterns on a thin catalytic ring are unaffected by end effects, and the corresponding periodic boundary conditions enable the simplest characterization and analysis. Mayer’s (1914) measurements of rotating electrical pulses in ring-shaped jellyfish tissue probably represents the first utilization of this advantage. Temperature patterns were observed during the oxidation of hydrogen on metallic and supported nickel catalysts (Graham et al., 1993a; Somani et al., 1996; Somani et al., 1997b) as well as carbon monoxide oxidation on metallic Pt (Yamamoto et al., 1995) and supported Pd rings (Liauw et al., 1997). Concentration patterns on a thin Pt ring were observed by Graham et al. (1994) during the oxidation of carbon monoxide. Figure 10 shows several typical temperature patterns on a catalytic ring as a function of time and the azimuthal position. These include a rotating hightemperature pulse (Figure 10a), a back-and-forth movement of two temperature fronts accompanied by intermittent formations and disappearances of cold spots in the high-temperature region (Figure 10b), antiphase oscillations of a high-temperature region between two locations (Figure 10c), and intermittent formations of a high-temperature region at two positions (Figure 10d). Figure 11a shows snapshots of a stationary, bounded high-temperature region observed during the oxidation of hydrogen at two different positions on a nickel-onalumina ring catalyst. The hot region could be placed at various locations on the ring by short, local preheating of different spots. This indicates that the evolution of the hot regions is not due to nonuniformity of the catalyst. The robust temperature patterns observed on catalytic surfaces under atmospheric conditions are strongly affected by global interaction (coupling) between the


Figure 10. Temperature patterns on a catalytic ring. (a) Rotating pulse during hydrogen oxidation on Ni. After Graham et al. (1993a). (b) Back-and-forth moving pulse during CO oxidation on a supported Pd ring catalyst. After Liauw et al. (1997). (c) and (d) Antiphase oscillations and intermittent pulse formation during H2 oxidation on Ni. After Somani et al. (1997b).

local reaction rate and the mixed reactants in the surrounding vessel. In that aspect, they differ from patterns formed by diffusion-reaction interaction that depend only on the local features of the state variables. The global interaction stabilizes pattern formation in some cases, while in others it tends to synchronize the behavior at various points. In a CSTR kept at a

constant temperature, the global coupling stabilizes the coexistence of regions with different temperatures for reactions with a monotonic rate dependence on the limiting reactant concentration. The reason is that any attempt at a high-temperature pulse to expand decreases the mixed reactants’ concentrations in the vessel. This, in turn, decreases the reaction rate at all


Figure 11. (a) Snapshots of a bounded high-temperature region at two different locations on the same supported nickel ring catalyst during the oxidation of hydrogen (96.5 vol %) by oxygen (3.5 vol %). The different states were generated by preheating different locations. (b) and (c) Local ring temperature, reactor temperature, and oxygen concentrations during a hot-spot formation following slow cooling of the reactor from 125 to 100 °C. After Somani et al. (1996).


Figure 12. A uniform-temperature state and a stationary temperature-front attained during ammonia oxidation on a Pt ribbon kept by electrical heating at an average temperature of 195 °C. After Lobban et al. (1989).

other points on the surface, inhibiting their ignition. Thus, the global interaction discourages the pulse expansion, which would have occurred in its absence. Figures 11b,c illustrate the role of the global interaction in the evolution and stabilization of a hightemperature region during hydrogen oxidation on a supported nickel ring catalyst in a mixed reactor while slowly decreasing its temperature. As the reactor and ring temperature decreased, the ambient oxygen concentration eventually increased to a level at which an ignited state could still exist at the lower reactor temperature. This led to the formation of a hightemperature region (hot spot) bounded by sharp temperature-fronts. Any expansion of the hot region decreases the oxygen concentration to a level at which the high-temperature state cannot exist any longer, and this arrests the attempted expansion. In the absence of the global interaction, a uniform ignited or extinguished state is attained. Experiments have shown that pattern evolution and stability may be affected by global interaction even for ultra-low-pressure reactions (Mertens et al., 1994). Probably the simplest example of pattern stabilization by global coupling in a reacting system is the formation of stationary or moving temperature patterns on catalytic ribbons heated electrically so that their total resistance or voltage remains constant. The first visual observation of this behavior was reported by Busch (1921). More recent observations were reported by (Barelko et al., 1978; Volodin et al., 1982; Sheintuch and Schmidt, 1986; Lobban et al., 1989; Philippou et al., 1991; Garske and Harold, 1992). Figure 12 describes a case in which either a uniform or nonuniform state exists during the oxidation of ammonia on a ribbon, the total resistance (average temperature) of which was kept constant. The steep temperature-front is stabilized by the electrical heating, as any expansion of the high(low-) temperature region increases (decreases) the wire average resistance and is counteracted by the electrical heating (Lobban et al., 1989; Sheintuch, 1989). Ignoring the potential temperature nonuniformities may lead to pitfalls in the analysis of kinetic data obtained by this convenient method. The dynamics of many nonreactive systems are affected by similar nonlocal variables; i.e., the local behavior depends also on some weighted integral of the states of all the elements in the system. To account for

Figure 13. Instantaneous oxygen concentration in a CSTR in which O2 reacts with hydrogen on a nickel ring. In case (a) a rotating temperature pulse existed on the ring, while uniform oscillations existed in case (b). The feed consisted of 5 vol % O2 and 80 vol % H2, the rest being nitrogen. Reactor temperature was 250 °C.

the influence of the global interaction, the diffusionreaction model, eq 16, has to be modified by adding an integral term that accounts for the nonlocal effect. Specifically, it has to be transformed to the form

∫

ut - D∇2u ) f(u) + g(u, h(u(x)) dx)

(17)

Elmer (1988, 1992) was the first to apply a model with global coupling to explain pattern formation in ferromagnetic resonance and ballast resistors. Similar patterns may form when an electrical current is passed through a superconducting wire (Gurevich and Mints, 1987). Purwins’ group (Willebrand et al., 1990) used a similar model to account for pattern formation in dc gas discharge. The interaction via global coupling among all the system elements enables formation of a rich class of patterns that would not form in its absence. This, in turn, increases the sensitivity of the system to the initial conditions; i.e., the larger number of possible patterns decreases the size of the regions of initial conditions leading to some of the states. Moreover, it affects pattern selection, specifically the one obtained under specific initial conditions. The global interaction also introduces novel types of bifurcations (transitions) among the qualitatively different states of the system (Middya et al., 1993, 1994; Graham et al., 1993b). The time-averaged conversion obtained when a pulse rotates around a ring may be either larger or smaller than that obtained by a uniform oscillatory state under the same operating conditions. One reason is that the ambient concentration is constant for a pulse rotating on a uniform medium, but it is time-dependent for an oscillatory reaction. Figure 13 describes a case in which a pulse rotating on a catalyst yields a much higher conversion than that on a uniform temperature-oscillating state under the same operating conditions. It would be useful to enhance our understanding of the class of reactions and operating conditions for which temperature pulses or other patterns may increase the conversion. Even more importantly, whether these patterns may increase the yield of a desired product over that attainable when the catalyst state is uniform, any potential application of such cases will require the


development of proper control policies for maintaining beneficial patterns. It is still an open question whether temperature patterns evolve on commercial catalytic pellets, which are usually smaller than those used in the experimental studies of the temperature patterns. Most catalytic surfaces operating under atmospheric conditions exhibit intrinsic and/or induced nonuniformities and anisotropies which may affect pattern formation and selection (Ba¨r et al., 1995; Schu¨tz et al., 1995; Kulka et al., 1995; Hagberg et al., 1996; Liauw et al., 1996; Bangia et al., 1996; Sheintuch, 1996). Obviously, nonuniformities may generate patterns which cannot form on uniform surfaces (Lobban et al., 1989; Philippou et al., 1991) and lead to interesting novel motions as well as pinning and reflection of fronts. Fascinating patterns may evolve when the nonuniformities regions with qualitatively different local phase-plane features exist on the catalyst surface (Liauw et al., 1996). The patterns generated by the nonuniformities are often rather similar to those formed by global interaction. Thus, it is rather difficult to determine whether an observed pattern is due either to nonuniformity, global interaction, or both. In addition, catalyst aging may also change the qualitative features of the temperature pattern (Graham et al., 1993a; Somani et al. 1997b). It will be important to account for the impact of nonuniformities and aging in any future attempts to exploit patterns to enhance reactor performance. Temperature-fronts evolve also in packed-bed reactors. Their dynamic features may be rather intricate, as the behavior at any point in the reactor is affected not only by its own features but also by those of other sections in the bed due to convection and dispersion. This communication with the up- and downstream locations may lead to temporal changes in the qualitative features of the local phase plane and to complex temperature patterns. Creeping temperature fronts in a packed-bed reactor were first reported by Frank-Kamenetskii (1955) and Wicke and Vortmeyer (1959) and are associated with a shift to a new state. The heat removed forms some particles by a downstream-creeping temperature front that preheats the fluid and causes the moving-front temperature to exceed that of a stationary front. The faster the front velocity, the larger the temperaturerise caused by the downstream movement. When the front moves in the upstream direction, its temperature is lower than that of a stationary front. Dvorak et al. (1994) have exploited the temperature-front movement to minimize the electrical power needed to preheat an automobile catalytic muffler. When the local dynamic features in a packed-bed reactor are either oscillatory or excitable, rather intricate temperature patterns may evolve in the reactor. For example, Puszynski and Hlavacek (1984) observed a periodic formation of a downstream-moving hot spot during the oxidation of carbon monoxide in a cooled packed-bed reactor (Figure 14). Wicke and Onken (1988) observed a similar periodic evolution of a moving hot front in an adiabatic reactor. Rovinsky and Menzinger (1993) carried out the oscillatory BelousovZhabotinsky reaction in a bed of particles on which ferroin was immobilized. This generated a periodic sequence of concentration pulses that moved downstream at a constant velocity (Figure 15). Sangalli and Chang (1994) have shown that differential convection and diffusion may generate similar as well as more

Figure 14. Periodic temperature waves in a packed-bed reactor in which CO is oxidized. After Puszynski and Hlavacek (1984).

Figure 15. Propagation of concentration waves in the part of a column (length of about 5.5 cm) in which the ferroin-catalyzed Belousov-Zhabotinsky reaction occurs. The light (dark) bands are of oxidized (reduced) ferroin regions. After Rovinsky and Menzinger (1993).

complex, irregular pulse motions in a packed-bed reactor under conditions so that a well-mixed reactor will not exhibit any instability. Barto and Sheintuch (1994) and Shvartsman and Sheintuch (1995) conducted a very comprehensive analysis of the dynamics of an adiabatic packed-bed reactor in which an exothermic reaction occurs on a catalyst which deactivates at high temperatures and is rejuvenated at low temperatures. In addition to previously reported motions such as periodic train of pulses and excitable behavior, they observed a rich class of novel motions. Figure 16 illustrates some of these interesting patterns found by Barto and Sheintuch (1994). Several of the intricate patterns were due to the existence of qualitatively different local phase-plane features in the bed and their temporal change by the behavior at other locations (Shvartsman and Sheintuch, 1994; Sheintuch and Shvartsman, 1994, 1996). These simulations suggest that the dynamic features of packed-bed reactors may be much more intricate than those observed and reported in the past. Important open questions are which of these predicted complex temperature patterns exist in commercial reactors, what is the impact that these patterns have on reactor performance, and what control policy can maintain beneficial patterns. Impact of Momentum Transport on Dynamics Most previous analyses of the dynamics of packedbed reactors ignored the interaction among the momentum transport, the chemical reactions, and the dependence of the physical properties of the fluid on


Figure 16. Simulation of temperature patterns in a packed-bed reactor. After Barto and Sheintuch (1994).

temperature and composition. However, this simplifying assumption may in some cases lead to drastic, qualitative differences between the model predictions and the behavior of the system. Multitube packed-bed reactors are widely used in the chemical industry to carry out highly exothermic reactions. It is generally assumed that the same flow rate exists in all the parallel tubes that are subjected to the same pressure-drop. However, in some cases, thermoflow multiplicity may occur; i.e., different flow rates exist in different parallel tubes due to the interaction among the momentum and energy balances and a sensitive dependence of the physical properties on the state of the fluid (concentration, temperature, etc.). Other systems that exhibit thermoflow multiplicity include flow in heated pipes (Pearson et al., 1973), evaporators (Ledinegg, 1938), and tubular reactors (Gupalo and Ryazantsev, 1968). Matros and Chumakova (1980) pointed out the possible development of thermoflow multiplicity in a packedbed reactor in which the reactants are incompressible. Lee et al. (1987, 1988) showed that the same development may occur if the fluid is compressible and that it is more likely to occur for reactions that involve a volume increase. Further analysis by Pita et al. (1989) showed that, in the presence of axial heat-conduction, thermoflow multiplicity may occur for a realistic set of parameters. This possible existence of different flow rates and temperature profiles in different tubes of a multitube or monolith reactor has to be taken into account in the design of these reactors. This thermoflow multiplicity illustrates the important role that the momentum transport may have on the reactor dynamics features. Hot spots have been reported to exist in many commercial reactors (Jaffe, 1976; Barkelew and Gambhir, 1984). They may lead to local deactivation of the catalyst and even anneal the reactor wall when present next to it, leading to an explosion. There are several possible causes for this undesirable phenomenon. One possible cause for the formation of hot spots is nonuniform packing of the reactor. This may occur especially with catalysts having irregular shapes and a wide particle-size distribution. Figure 17 describes hot spots formed in a fixed-bed reactor during isobutyl alcohol oxidation. The characteristic size of the “hot

Figure 17. Isotherms at the exit of a 120-mm long bed of copper oxide catalyst in which isobutyl alcohol is oxidized. After Matros (1985).

spots” was a 10-15 particle diameter. Another possible cause is incomplete coke-burning during regeneration, which may generate nonuniform voidage-distribution in the bed, which will, in turn, lead to hot-spot formation. Another intriguing mechanism is the instability of the axially symmetric solutions, i.e., hot-spot evolution due to symmetry-breaking. The mathematical analysis and simulations of these three-dimensional systems is obviously rather intricate. Nguyen and Balakotaiah (1994) investigated this possible mechanism of hot-spot formation. They concluded that, in the case of slow liquid flow rates (residence time of the order of minutes), isolated branches of states with very high catalyst temperature may exist. In general, these states cannot be reached without a large perturbation. The nonuniform flow caused by the variation in the physical parameters with temperature perturbs the local transport coefficients. These perturbations are sometimes sufficiently large to shift the system locally to a very high-temperature state, i.e., form a hot spot. Despite its important safety implications, our ability to predict the evolution, size, and location of these hot spots in commercial reactors is still in its infancy. There is a definite need to enhance our understanding of this phenomenon.


Concluding Remarks

Subscripts

The movement of temperature-fronts during the wrong-way behavior and the RFO of packed-bed reactors is reasonably well understood at present for the singlereaction case. There is still a need to improve our understanding of the front dynamics when several reactions occur simultaneously and when reversible changes occur in the state of the catalyst (such as adsorption-desorption of the reactants on an inert support) which do not affect the steady-state behavior, but which can affect the transient behavior. Many open questions still remain concerning the formation of temperature patterns on single and arrays of catalytic pellets and in packed-bed reactors. In particular, it is important to determine if and for which commercial processes these patterns may exist, what their impact is, and what control policies can maintain a beneficial temperature pattern in the presence of intrinsic and induced catalytic surface nonuniformities and aging. Our ability to predict several important dynamic features in packed-bed reactors, such as the formation and location of hot spots, will require the analysis of dynamics models, which account also for the momentum balance and variation in the physical properties in the reactor. Hopefully, answers to many of these challenging and important open questions will be forthcoming in the near future.

e ) effective f ) feed conditions g ) gas phase i ) inhibitor max ) maximum temperature s ) solid phase sg ) solid-gas 0 ) feed

Acknowledgment I am thankful to the NSF and the Welch Foundation for support of my research in this area and to M. A. Liauw for helpful comments. I am grateful to the organizing committee for the invitation to participate in the celebration of Prof. G. Froment’s 65th birthday, which led to the writing of this review. It is a pleasure to acknowledge the hospitality of the Fritz-Haber Institut, Max-Planck Gesellschaft, Berlin, where this manuscript was written. Nomenclature av ) interfacial area per unit volume C ) concentration of reactant cp ) specific heat D ) diffusion coefficient E ) activation energy h ) heat-transfer coefficient kˆ (T) ) rate constant at T kˆ ) pre-exponential factor of rate constant kc ) mass-transfer coefficient Le ) Lewis number, defined by eq 4 n ) reaction order NR ) quantity defined by eq 10 R ) universal gas constant t ) time T ) temperature Tc ) critical temperature, defined by eq 12 u ) velocity w ) dimensionless velocity, defined by eq 4 z ) distance from reactor inlet Greek Symbols λ ) thermal dispersion ) void fraction of bed F ) density -∆H ) heat of reaction ∆Tad ) adiabatic temperature-rise, defined by eq 10

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Received for review September 30, 1996 Revised manuscript received February 4, 1997 Accepted February 4, 1997X IE960597K

X Abstract published in Advance ACS Abstracts, June 15, 1997.

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