Effect of the Switch Strategy on the Stability of Reactor Networks

Effect of the Switch Strategy on the Stability of Reactor Networks

Erasmo Mancusi

Facolta d’ Ingegneria, UniVersita del Sannio, Piazza Roma , 82100, BeneVento, Italy

Lucia Russo,* Pietro Altimari, Pier Luca Maffettone, and Silvestro Crescitelli

Dipartimento d’Ingegneria Chimica UniVersita “Federico II” Piazzale Tecchio 80, 80125 Napoli, Italy

The dynamics of a three-catalytic-reactor network, where an exothermic irreversible reaction occurs, is studied.The network is operated by periodically switching feed and discharge positions (with the period beingT).We analyzed the effect of two different switch strategies and observed different spatio-temporal patterns.Such differences result from the different spatio-temporal symmetry that is induced by the switch strategies.Their effect on the stability is studied by comparing the stability range ofT-periodic regimes as the switchtime is varied for different values of the feed temperature. This analysis is conducted by a numerical continuationtechnique based on the spatio-temporal symmetry of the forced system. For low (ambient) feed temperature,the stability ranges of a diabatic reactor conform to criteria previously proposed for adiabatic reactors [Sheintuch,M; Nekhamkina, O.AIChE J. 2005, 51, 224]. We observe that these criteria cannot be applied at higher feedtemperatures, as a consequence of the external heat losses. One of the examined switch strategies is used todetermine a significant widening of the operability conditions.

Introduction

Several studies have shown that many irreversible exothermiccatalytic processes can be efficiently performed in periodicallyforced fixed-bed reactors.2-5 The reverse-flow reactor (RFR),which is a fixed-bed reactor with a periodical inversion of theflow direction (as originally proposed by Cottrell6) is, by far,the most extensively studied periodically forced reactor.

In RFRs, the flow inversion traps the reaction heat in thecentral part of the reactor, with the edges remaining colder.Under optimal conditions, after an initial transient period, a bell-shaped temperature profile is attained and cold and very leanmixtures can be conveniently processed.7 RFRs have beenproven to be very efficient to process pollutant mixtures thathave low adiabatic rise (see, for example, refs 8-12). Indeed,many experimental and theoretical studies have shown thatautothermal catalytic combustion of lean volatile organiccompound (VOC) mixtures may be conducted in an RFR witha significant reduction of energetic costs (see the work of Kolioset al.13 for a review).

RFRs have also been successfully applied to other catalyticprocesses that involve equilibrium-limited exothermic reactions14

or the coupling of endothermic and exothermic reactions.15 Thecapability of fixed-bed reactors to develop reaction fronts canbe exploited in RFRs to generate spatio-temporal patterns thatenable not only the recovery of reaction heat but also theoptimization of the reaction path.

The main shortcoming of RFRs is the loss of a fraction ofreactants immediately upon flow reversal: this is the so-calledwashout effect. Matros2 has explored several other reactorconfigurations where the objective has been to trap reaction heatinside the catalytic bed and avoid washout. One proposedconfiguration is a network (abbreviated hereafter as NTW) oftwo or three reactors with feed and discharge positions periodi-cally shifted in time. In the NTW, each reactor is usedcontinuously. However, other possibilities have been proposed.

For example, for the catalytic combustion of methane, Balajiand Lakshminarayanan16 have proposed switching strategieswhere only a selected number of reactors are used at eachswitching; for the synthesis of methanol, a different reactorconfigurationscalled the “star” reactor configurationswasproposed.17

In this paper, we analyze the case of NTWs, because severalstudies have confirmed that networks of two or more catalyticbed reactors can be a promising alternative to RFRs.18,19 ForNTWs, the periodic regime is induced by periodically varyingthe feed position according to a cyclic permutation of the reactorsequence. The heat front propagates in a virtually closed cyclewhile keeping the flow direction in each reactor constant.Haynes and Caram18 compared an RFR with a NTW of tworeactors for catalytic combustion of VOCs, and they noted adrastic reduction of washout and more-uniform use of thecatalyst. Fissore and Barresi,20 by comparing the performancesof a NTW of three reactors with a RFR, showed that themaximum temperature attained in the NTW is higher, whereasthe stability range is narrower. The application of NTW toequilibrium-limited exothermic reactions seems promising. Forexample, in the case of methanol synthesis, the NTW canachieve better conversion, with respect to the RFR.21

From the theoretical point of view, Sheintuch and Nekhamki-na1,22 analyzed a NTW that consisted of an arbitrary number ofreactors, and showed that its performance converges to that ofa continuum loop reactor as the number of reactors is increased.

Generally, the optimal regimes in the periodically forcedreactor NTW are periodic with the same period of the forcing:if the period of the forcing is denoted asT, such a regime isT-periodic. Usually, such a regime is characterized by a reactionfront traveling along the reactors. However, under certainoperating conditions, much-richer dynamics with multiperiodic,quasi-periodic, and chaotic spatiotemporal-patterns was pre-dicted for NTWs23,24 and RFRs.25

When the NTW consists of fixed-bed reactors, the periodicforcing generates spatio-temporal patterns that are strictly relatedto the spatio-temporal symmetry of the system.23,26 Indeed, itwas demonstrated that periodic forcing of a NTW of identical

* To whom the correspondence should be addressed. Tel.:+39 081768262. Fax:+39 081 7682262. E-mail address: [email protected].

6510 Ind. Eng. Chem. Res.2007,46, 6510-6521

10.1021/ie0701266 CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 09/01/2007

reactors determines specific spatio-temporal symmetries27 thatare not dependent on the specific model used to describe thereactor. It was also shown that the symmetry properties of thesystem have substantial effects on the bifurcation behavior ofthe NTW.

In the present paper, we consider a NTW of three fixed-bedreactors forced with two different switch strategies characterizedby different feed/discharge permutations. The objective of theanalysis is to evaluate how the switch strategy affects the NTWperformances. The first strategy, which was already consideredin the literature, consists of changing at each switch time thefeed/discharge positions, so that the first reactor of the NTW ismoved to the last place, whereas with the second strategy, thelast reactor is periodically moved to the first place of the NTWsequence.

We discuss the differences in the spatio-temporal symmetryinduced by the two different strategies, as well as how thissymmetry can be recognized in the generated spatio-temporalpatterns. We show that different forcing strategies induce verydifferent stability ranges of theT-periodic regimes in the NTW.This is demonstrated both by bifurcation analysis and bycomparison of the related spatio-temporal patterns generatedby the two strategies. The bifurcation analysis is performed bychoosing the switch time and the feed temperature as bifurcationparameters. The stability limits are then compared with thosepredicted by the criterium proposed by Sheintuch and Nekhamki-na.1

2. Mathematical Model Equations and Switch Strategies

We consider a NTW of three identical fixed-bed reactorswhere an exothermic irreversible catalytic reaction occurs. Asan example, the combustion of VOCs over a catalytic bed isconsidered. Two different switch strategies, as sketched in Figure1 , are studied. In both cases, the sequence of reactors is changedperiodically following a cyclic permutation.

For strategy 1, the feed and discharge positions moveclockwise, and the reactors are fed according to the sequence1-2-3 in the time range [0,τ[ (where τ is the switch time);after the first switch (that is, in the range [τ, 2τ[), they are fedaccording to the sequence 2-3-1; then, after the second switch(t ∈ [2τ, 3τ[), the reactors are fed as the sequence 3-1-2; thenext switch brings the feed sequence back to that of the firstone (that is 1-2-3), and the permutation cycle restarts. Thus,the forcing period is defined asT ) 3τ.

For strategy 2, the feed and discharge positions movecounterclockwise (see Figure 1). In fact, after the first switch

(that is, in the range [τ, 2τ[), the fresh feed enters reactor 3 andthe discharge happens through reactor 1 (sequence 3-1-2);then, after the second switch (t ∈ [ 2τ, 3τ[), the reactors are fedas the sequence 2-3-1; with the next switch, the feed sequenceis again 1-2-3, and the permutation cycle restarts.

2.1. Model Equations.We consider a first-order exothermicreaction that occurs in a fixed catalytic reactor. Each fixed-bedreactor is modeled as a heterogeneous system with heat- andmass-transfer resistance between the gas phase and the solidphase, axial dispersion in the gas phase, axial heat conductionin the solid phase, and cooling at the reactor wall.

The mathematical model23 for the NTW reads

In eq 1, the first two equations represent the mass balance inthe gas phase and in the solid phase, respectively. The thirdand fourth equations are the heat balance in the gas phase andin the solid phase, respectively. The indexi identifies the reactor.Equations 1 are dimensionless, with an axial coordinate definedasz ) x/L, time defined ast ) Vth/L, the gas-phase conversiondefined asyg ) (C0 - Cg)/C0, the conversion on the catalystgiven asys ) (C0 - Cs)/C0, the gas-phase temperature definedasθg ) E(T - T0)/RT0

2, and the catalyst temperature given asθs ) E(Ts - T0)/RT0

2. The definition of all dimensionlessparameters and their values are reported in Table 1. Theremaining relevant symbols are explained in the nomenclaturesection at the end of this paper.

The forcing enters the model through the boundary conditions,which, hence, differentiate the two strategies. The forcing isimplemented with a discontinuous periodic wave functionf(t)(see Figure 2), given by

Thus, the boundary conditions for strategy 1 reads

where

Figure 1. Depiction of three catalytic reactors with two different strategiesof permutation of the feed and discharge positions. {∂yg,i

∂t) 1

Pegm

∂2yg,i

∂z2-

∂yg,i

∂z+ Jg

m(ys,i - yg,i)

Jsm(ys,i - yg,i) ) ηDa(1 - ys,i) exp[ θs,i

1 + (θs,i/γ)]∂θg,i

∂t) 1

Pegh

∂2θg,i

∂z2-

∂θg,i

∂z+ Jh

g(θs,i - θg,i) - æ(θg,i - θw,i)

∂θs,i

∂t) 1

Pesm

∂2θs,i

∂z2- Js

h(θs,i - θg,i) + BηDa(1 - ys,i) exp[ θs,i

1 + (θs,i/γ)] }(for i ) 1, 2, 3) (1)

f(t) ) {0 if 0 etτ

mod 3< 1

1 iftτ

mod 3> 1 } (2)

{∂yg,i

∂z |0)

- [1 - f(t - (i - 1)τ)]yin - f(t - (i - 1)τ)yg,i-1(1,t) + yg,i(0,t)

Pegm

∂θg,i

∂z |0)

- [1 - f(t - (i - 1)τ)]θin - f(t - (i - 1)τ)θg,i-1(1,t) + θg,i(0,t)

Pegh

∂θs,i

∂z |0) 0

}(3a)

∂yg,i

∂z |1

)∂θg,i

∂z |1

)∂θs,i

∂z |1

) 0

Ind. Eng. Chem. Res., Vol. 46, No. 20, 20076511

while, for strategy 2,

where

Note that the NTW mathematical model is a discontinuousperiodically forced system with a forcing period ofT ) 3τ forboth switching strategies.

For subsequent numerical investigation, the infinite dimen-sional PDE system (eqs 1-3) has been reduced to a set of 108ordinary differential equations (ODEs) by an orthogonal col-location technique on finite elements.28 The domain of eachreactor has been divided in three blocks, and four collocationpoints have been used in each subdomain.

2.2. Symmetry Properties.The mathematical model (eqs1-3) that are used to describe the reactor network can be writtenin abstract form as

In eq 4,x ≡ (x1,x2,x3) is the state vector of the system;xi ≡(yg,i,θg,i,θs,i) is the state of each reactor, which is representedby the spatial temperatures and gas conversion profiles, whichare functions of variablest andz. The vectorλ is the parametervector, which includes all the parameters of the model (see Table1).

The difference in the forcing term gives a different spatio-temporal symmetry of the model equations. For strategy 1,23

the spatio-temporal symmetry can be expressed as

with

where I is the identity transformationIxi ) xi and G acts onthe state vectorx as follows:

The groupΓ ) {I,G,G2} is associated to the space transforma-tions, and the group∆ is associated to time translations ofkτ,wherek a positive integer.29 For each transformationγ ∈ Γ,there exists a time translationδ ∈ ∆, such that the vector fieldis invariant under the action ofγ X δ.30,31Thus, the set of suchcouples (γ,δ) ∈ Γ × ∆ defines the spatio-temporal symmetryfor strategy 1, which is homeomorphic toZ3×S1.29

Similarly, for strategy 2, it is apparent that

with

whereG′ acts as follows:

In this case, the group that is associated with the spacetransformation isΓ′ ) {I,G′,G′2}. This strategy creates spatio-temporal symmetries (γ,δ) ∈ Γ′ × ∆, which is homeomorphic

Table 1. Parameter Values in eqs 1-3a

parameter symbol/expression value

Damkohler number Da ) Lr in/(Vεs) 0.32effectiveness factor of the catalyst η 1.nondimensional activation energy γ ) E/(RT*0) 16.68Peclet number for mass in gas phase Peg

m ) LV/D 105.82Peclet number for energy in gas phase Peg

h ) FgcpgLV/kg 214.66Peclet number for energy in solid phase Pes

m ) FscpsLV/ks 0.48× 105

Lewis number (according to ref 1) σ ) [Fgcpgε + (1 - ε)Fscps]/Fgcpgε 1281nondimensional mass-transfer coefficient Jg

m ) kmaL/(Vε) 5.67nondimensional mass-transfer coefficient Js

m ) kmaL/[Vεs(1 - ε)] 7.63nondimensional heat-transfer coefficient Jg

h ) hfcaL/(FgcpgVε) 9.46nondimensional heat-transfer coefficient Js

h ) hfcaL/[FscpsV(1 - ε)] 7.5× 10-3

nondimensional adiabatic temperature increase B ) ∆HCA,inγ/(FgcpgT*0) 0.0097nondimensional inlet conversion yg,in 0.nondimensional wall temperature θw ) (T*w - T*0)γ/T*0 -9.6nondimensional heat-transfer coefficient at the wall φ ) 4hfwL/(FgcpgdrVε) 0.24inlet temperature Tin 25-300°Cswitch time τ 0-20000

a Characteristic values of variables: inlet concentration,Cin ) 0.003 kmol/m3; V ) 2 m/s;L ) 0.17 m; andT0 ) 730 K.25

Figure 2. Forcing function.

{∂yg,i

∂z |0)

- [1 - f(t + (i - 1)τ)]yin - f(t + (i - 1)τ)yg,i-1(1,t) + yg,i(0,t)

Pegm

∂θg,i

∂z |0)

- [1 - f(t + (i - 1)τ)]θin - f(t + (i - 1)τ)θg,i-1(1,t) + θg,i(0,t)

Pegh

∂θs,i

∂z |0) 0

}(3b)

∂yg,i

∂z |1

)∂θg,i

∂z |1

)∂θs,i

∂z |1

) 0

{x3 ) F(x,λ,t) F(x,λ,t) ) F(x,λ,t + T)x(t0) ) x0

} (4)

GF(x,λ,t) ) F(Gx,λ,t - τ) (5)

G ) (0 I 00 0 II 0 0)

G(x1,x2,x3) ) (x2,x3,x1) (6a)

G2(x1,x2,x3) ) (x3,x1,x2) (6b)

G3 ) I (6c)

G′F(x,λ,t) ) F(G′x,λ,t - τ) (7)

G′ ) (0 0 II 0 00 I 0)

G′(x1,x2,x3) ) (x3,x1,x2)

G′2(x1,x2,x3) ) (x2,x3,x1)

G′3 ) I (8)

6512 Ind. Eng. Chem. Res., Vol. 46, No. 20, 2007

to Z3 × S1 as well. Thus, the two strategies differ for thegeneratorsG andG′.

The difference in the spatio-temporal symmetry is reflectedin differences between the spatio-temporal patterns of the twostrategies. In fact, in the case of theT-periodic regimes of interestin this paper, the spatio-temporal symmetry of the regimes forstrategy 1 implies that

whereas, for strategy 2,

Thus, from eqs 9 and 10, for both strategies, theseT-periodicregimes seem to be discrete traveling (or rotating) waves(Sheintuch and Nekhamkina1 called these regimes “rotating-pulse” regimes).

3. Effect of Switch Strategy

The effect of switch strategy on the dynamic behavior of theNTW has been studied through spatio-temporal patterns andbifurcation analysis, in the case ofT-periodic regimes. Theswitch time and the inlet temperature are chosen as bifurcationparameters, because they are key parameters for design, opera-tion, and control. In fact, the switch time greatly affects therange of stability of the NTW,20 whereas the feed temperaturemight vary significantly, possibly causing the NTW extinction.26

Bifurcation analysis is performed with a continuation algo-rithm that allows the construction of solution diagrams thatrepresent the locus of theT-periodic regime solution. Weimplemented this procedure as described in detail by Russo etal.;27 this technique exploits symmetry properties (induced bythe periodic forcing) of the model. Stability ranges of ignitedT-periodic regimes and their bifurcations are thoroughly ana-lyzed as the bifurcating parameters are varied for both switchingstrategies. In such a way, ignition/extinction ranges of the forcednetwork are efficiently detected.

3.1. Effect of Switch Strategy on Spatio-temporal Patterns.The occurrence of temperature and conversion waves travelingalong the NTW appears asT-periodic regimes.

An example for strategy 1 of a traveling temperature waveis shown in Figure 3, where spatial profiles at three successivetimes within a switch interval are reported. This wave ischaracterized by two different velocities indicated withV1 and

V2 in Figure 3: V1 is the velocity with which the cold regiondownstream from the maximum temperature moves toward theNTW exit; V2 is the velocity with which the hot region upstreamfrom the maximum temperature moves toward the NTW exit.

A more-complete picture of the same traveling wave is givenby the spatio-temporal pattern reported in Figure 4, where eachpanel represents a single reactor. The temperature level is codedin shades of gray, and the nondimensional axial coordinate isreported as the abscissa. The scaled time (t/τ) is given as theordinate. For the sake of clarity, only two temperature levelsare considered: light gray represents points withT > 500 °C,and dark gray representsT < 500 °C (thus, from this pointforward, we considerV1 andV2 to be equal to the velocities ofthe point at 500°C downstream and upstream from themaximum temperature, respectively). In Figure 4, the path ofthe temperature traveling wave through the NTW is apparent:the temperature front forms in the first reactor, and then movesalong axially; after the first switch, a new front forms and movesalong the second reactor; then, after the second switch, the samehappens in the third reactor. It is evident that the spatio-temporalpattern (Figure 4) in any of the three reactors is equal to thespatio-temporal pattern in the successive (according the flowdirection) reactor but shifted forward in time byτ. This featureis the manifestation of the spatio-temporal symmetry expressedby eq 9.

A typical spatio-temporal pattern of aT-periodic wave forstrategy 2 is reported in Figure 5.

The differences in the spatio-temporal symmetries inducedby the two forcing strategies are readily evident from thecomparison of the spatio-temporal patterns of theT-periodicregime (see Figures 4 and 5). The situation for strategy 2 (Figure5) shows that the spatio-temporal pattern in any of the threereactors is equal to the spatio-temporal pattern in the preceding(according the flow direction) reactor but shifted forward in timeby τ, or, equivalently, equal to the spatio-temporal pattern inthe successive (according the flow direction) reactor but shiftedbackward in time byτ, which corresponds to the propertiesexpressed in eq 10.

Note that the spatio-temporal patterns reported in Figures 4and 5 can be used to estimateV1 andV2. In fact, the slope ofthe boundary line separating the cold region from the hot regiongives an estimation ofV1, whereas the slope of the boundaryline separating the hot region from the cold one is an estimationof V2. These velocitiesV1 and V2 are strictly related to twocharacteristic velocities of temperature fronts in fixed beds:Vth,which is the velocity of a thermal front, in the absence ofreaction, andVfr, the velocity of a reaction front that representsthe velocity at which the reaction section moves along thereactor.

V1 can assume two values: it is equal toVfr when the frontcrosses a bed region where a chemical reaction is occurring, orit is equal toVth when the front moves in a region of the reactorwhere the reaction is not occurring. In contrast,V2 is alwaysequal toVth. Consequently,Vth coincides with the inverse ofthe slope of the lower boundary line separating the hot regionfrom the cold region.

It is well-known that the nondimensional velocity of thethermal front (Vth) is the ratio between the gas heat capacityand the effective heat capacity;32,33 thus, it is the reciprocal ofσ. As far as the velocity of the reaction front is concerned,because of the complexity of the mathematical model (eqs 1-3),it is not possible to analytically evaluateVfr as done for pseudo-homogeneous models (for the case of an adiabatic reactor,34 orfor the case of a non-adiabatic reactor).35 It turned out, however,

Figure 3. Spatial gas temperature profile for strategy 1, along the network(NTW) at three successive time instants;t ) 1500,t ) 2500, andt ) 3000.The inlet temperature isTin ) 60 °C and the switch time isτ ) 3000.

x1(z,t) ) x2(z,t + τ) ) x3(z,t + 2τ) (9)

x1(z,t) ) x3(z,t + τ) ) x2(z,t + 2τ) (10)


that Vth andVfr are slightly affected by the switch strategy, asthey are mainly dependent on the heat capacities of solid andgas, on the gas velocity, on the adiabatic temperature rise, andon the inlet temperature,32,33 and both velocities (Vthand Vfr)can be estimated by the analysis of the spatio-temporal patternsthrough the evaluation ofV1 andV2.

3.2. Bifurcation Analysis: The Influence of Switch Time(τ). Figure 6 shows the solution diagram for strategy 1 withthe switch timeτ as the bifurcation parameter. In this solutiondiagram, the locus ofT-periodic regimes is shown. Stablesolutions are represented by solid lines, and unstable solutionsare plotted as dashed lines. The feed temperature value is 60°C, and the other parameter values are those reported in Table1. Note that, at this feed temperature value, the unforced systemdoes not exhibit any stable ignited regime.

A nonignited T-periodic regime is present throughout theinvestigated switch time range (the horizontal line at the bottomof Figure 6). The ignited (high conversion) solutions form anisola bounded by two catastrophic36 saddle node bifurcationpoints: S1 (at τ ) 1386) and S2 (at τ ) 3950). As the switchtime is varied within the rangeτS1 < τ < τS2, stable highconversionT-periodic regimes coexist with stable nonignitedT-periodic regimes.

When the switch time is increased aboveτS2, the forcing isnot able to sustain autothermal operation anymore, and, cor-respondingly, no high conversion solution is found. Conversely,for switch times lower thanτS1, T-periodic ignited solutionsdisappear but the system still possesses ignited solution in therangeτ ∈ [∼1, 1386], which are stable multiperiodic, quasi-

periodic, and chaotic regimes. Those solutions can be detectedvia numerical simulation. A detailed description of the bifurca-tions that occur in this switch time range is beyond the scopeof this work (for details, see the work by Russo et al.23).

Figure 7 shows the solution diagram ofT-periodic regimesin the case of switch strategy 2. This solution diagram wasobtained with the same operating conditions used for strategy1 (see Figure 6).

Figure 4. Spatio-temporal gray-scale patterns of the gas-phase temperature of aT-periodic regime for strategy 1 forTin ) 60 °C andτ ) 2500. Each boxcorresponds to a reactor unit.

Figure 5. Spatio-temporal gray-scale patterns of the gas-phase temperature of aT-periodic regime for strategy 2 forTin ) 60 °C andτ ) 5000. Each boxcorresponds to a reactor unit.

Figure 6. SymmetricT-periodic solution diagram with the switch time (τ)as the bifurcation parameter for strategy 1 atTin ) 60 °C. The solution isreported with the gas temperature at the network exit (Tg,out, given in Celsiusdegrees). Solid lines represent stableT-periodic regimes; dashed linesrepresent unstableT-periodic regimes.


Again, a nonignitedT-periodic regime is present throughoutthe investigated switch time range. The ignited (high conversion)solutions form an isola bounded by two saddle node bifurcationpoints S′1 (at τ ) 2810) and S′2 (at τ ) 7900). As the switchtime is varied within the rangeτN-S < τ < τS′2

, stable highconversion T-periodic regimes coexist with nonignitedT-periodic regimes. In this case, the stable high conversionT-periodic regimes on the isola become unstable atτ ) 2870through a Neimark-Sacker bifurcation N-S. This bifurcationis supercritical, and leads to the onset of stable symmetric quasi-

periodic regimes.36 It is important to note the switch time valueat which the Neimark-Sacker occurs is very close to the switchtime value at which the saddle-node bifurcation S′1 occurs.Thus, only the saddle-node bifurcation S′2 is catastrophic.

As for the case of strategy 1, as well as strategy 2 for switchtimes lower thanτS′1

in the rangeτ ∈ [∼1, 2790], stablemultiperiodic, quasi-periodic, and chaotic regimes are detected.

The most important feature of the solution diagram shownin Figure 7 is the significantly larger stability range ofT-periodicignited solutions, with respect to that predicted for strategy 1

Figure 7. SymmetricT-periodic solution diagram with the switch time (τ) as the bifurcation parameter for strategy 2 atTin ) 60 °C. The solution is reportedwith the gas temperature at the network exit (Tg,out, given in Celsius degrees). A Neimark-Sacker bifurcation is indicated with the filled square; the othersymbols are the same as those adopted in Figure 6.

Figure 8. Spatio-temporal gray-scale patterns of the temperature in the gas phase of aT-periodic regime: (a) strategy 1 and a switch time ofτ ) 3950; (b)strategy 2 and a switch time ofτ ) 7900. Each box corresponds to a reactor unit. The feed temperature in the gas phase isTin ) 60 °C.


(see Figure 6). In particular, the switch time range (τ ∈[2810, 7900]), where strategy 2 exhibits stable traveling waves,is almost doubled, with respect to that observed for strategy 1(τ ∈ [1386, 3950]). This fact seems to suggest that an intelligentchoice of the switch strategy can overcome the stabilitylimitations of NTW reported in the literature.20

A physical interpretation of these limits is possible using theanalytic criteria of Sheintuch and Nekhamkina,1 who proposedand tested these criteria for a NTW of adiabatic reactors treatedwith a pseudo-homogeneous model. These criteria define therange of existence of traveling waves, and they are based onthe comparison of the thermal and reaction front velocity withthe switch velocity (that is, the velocity of the feed positionmovement). The range of existence of aT-periodic ignitedsolution is defined by the following inequalities:

whereVsw is the switch velocity. As discussed previously,Vth

andVfr are not significantly dependent on the switch strategy.In contrast, the switch velocity is dependent on the switchstrategy, asVsw ) n/τ, wheren is the number of the reactorunits jumped upon switching (in the flow direction) by the feedposition. It can be easily noted that the value ofn is 1 in thecase of strategy 1, and 2 in the case of strategy 2.

Thus, eq 11 can be reformulated as follows:

For the calculations reported in Figures 6 and 7,σ ) 1281andVfr ≈ 2.57× 10-4; consequently, the range of existence ofT-periodic solutions predicted through eq 12 is 1280e τ e

3890 for the strategy 1, and 2560e τ e 7780 for strategy 2.These ranges quantitatively agree with the solution diagrams(Figures 6 and 7), even though they were obtained undernonadiabatic conditions and with a heterogeneous model. Thus,it is apparent that, in passing from strategy 1 to strategy 2, adoubling of the operability limits is observed via continuationand predicted by eq 12.

Indeed, the upper existence limit in eq 12 imposes that theswitch time must be lower thann/Vfr to avoid the reaction frontexit from the NTW, and, consequently, the reactor extinction.This situation is apparent in Figure 8, where the spatio-temporalpatterns of the gas-phase temperature for both strategies arereported close to extinction conditions. As predicted by eq 12,the extinction condition for strategy 1 is attained when the switchtime coincides with the time needed for the reaction front totravel along one reactor (Figure 8a), whereas, for strategy 2,extinction occurs when the switch time coincides with the timeneeded by the reaction front to cover two reactors (Figure 8b).It should also be noted that, under the upper extinction conditionVsw ) Vfr, V1 is constant and equal to the reaction front velocity(Vfr). In fact, for both strategies, in the spatio-temporal-patternsin Figure 8, the hot region is delimited by two straight lines.

Regarding the lower limit conditions, in the case underinvestigation, they represent extinction conditions ofT-periodicregimes, even though other ignited regimes can also be foundat lower switch times.

The spatio-temporal patterns for a switch time close toσ arereported in Figure 9. Significant differences are found whencomparing the lower and upper extinction conditions for bothswitch strategies. The most important difference is the noncon-stant value ofV1, in the case of the lower extinction limit. Theline separating the cold region from the hot region, downstreamfrom the maximum temperature, shows three different slopes.

Figure 9. Spatio-temporal gray-scale patterns of the temperature in the gas phase of aT-periodic regime: (a) strategy 1 and a switch time ofτ ) 1400; (b)strategy 2 and the switch time isτ)2890. Each box corresponds to a reactor unit. The feed temperature in the gas phase isTin ) 60 °C.

Vfr e Vsw e Vth (11)

nσ e τ en

Vfr(12)


The almost horizontal slope is linked to the development ofthe front upon switching; the largest slope corresponds toVfr,which is then followed by a faster velocity (smaller slope) equalto Vth. This scenario is completely different from that appearingclose to upper extinction limit (Figure 8) where constant valuesof V1 were found throughout the NTW. Indeed, it is instructiveto analyze the first reactor in Figure 9a for strategy 1. Duringthe first time interval [0,τ], after a space lag that is required toform the front, the reactor is fed with fresh feed andV1 is equalto the reaction front velocity (V1 ) Vfr). Upon switching, thefirst reactor is fed with an exhausted stream exiting from reactor3, and, thus,V1 ) Vth. A similar discussion holds also for thecase of strategy 2 in Figure 9b.

3.3. Bifurcation Analysis: The Influence of Feed Tem-perature (Tin). So far, we have observed that the range ofexistence of ignitedT-periodic traveling wave solutions can beadequately predicted with eq 12 for both switch strategies undernonadiabatic conditions. A consequence of eq 12 is the doublingof limiting conditions upon changing from strategy 1 to strategy2. We have analyzed the adiabatic operations via continuation,and we have observed that such a doubling is perfectly fulfilledin that case for different feed temperatures. On the other hand,under nonadiabatic conditions, the doubling is not rigorouslysatisfied. In this section, we will show that, as the feedtemperature increases, the observed existence range for theT-periodic traveling wave does not double when passing fromstrategy 1 to strategy 2.

Figure 10 shows theT-periodic solution diagrams as theswitch timeτ is varied for strategy 1 (panel a) and strategy 2(panel b), calculated at a larger feed temperature (Tin ) 240°C) under nonadiabatic conditions (see parameterφ in Table1); this solution diagram can be compared with those reportedin Figures 6 and 7 for a lower inlet temperature. For bothstrategies, two isolas are now observed: the isola at larger switchvalues corresponds to that observed at lower feed temperatures,because it is characterized by traveling wave solutions; a newisola is observed at lower switch times, but in this case, as itwill be addressed below, theT-periodic solutions are nottraveling waves.

The isola for large switch time values identifies the range ofexistence of theT-periodic traveling waves. The lower limit(τS1 ) 1390 andτS′1

) 2710, respectively) is, again,nσ, thusfulfilling the predicted doubling. However, the upper limitbehaves differently, because, for strategy 1,T-periodic regimes

vanish atτS2 ) 8020, but, for strategy 2,τS′2) 14 100, which

is a value that is significantly less than 2τS2.Indeed, the dependence onTin of the saddle-node bifurcations

S2 (see Figures 6 and 10a) and S2′ (see Figures 7 and 10b),which mark the higher limit ofT-periodic traveling wavesolutions for the two strategies, can be determined with twoparameter continuation withTin and τ as the bifurcationparameters.

Figure 11 shows the loci of the two saddle node points (S2

and S′2) in theTin-t′ plane for both switch strategies. Note that,to compare the two strategies more easily, we used a normalizedswitch timet′ (t′ ) τ/n), so that its value is strategy-independent.It is apparent that, at low inlet temperatures, the two curves inFigure 11 overlap, thus confirming the doubling of the limitingconditions as the strategy changes from strategy 1 to strategy2. AsTin increases, the two curves separate, showing a reduction,with respect to the predicted factor of 2. As the switch timeincreases, the feed temperature needed to sustain the autothermalcombustion of VOCs eventually approaches the same asymptoticvalue for both strategies. This value coincides with the ignitiontemperature of the unforced network of three reactors (∼280°C26).

The significant mismatch, with respect to the prediction of adoubling of the upper extinction limit, can be rationalized as

Figure 10. SymmetricT-periodic regimes solution diagram with the switch timeτ as the bifurcation parameter atTin ) 240 °C: (a) strategy 1 and (b)strategy 2. The solution is reported with the gas temperature at the network exit,Tg,out (given in Celsius). A Neimark-Sacker bifurcation is indicated withthe filled square, and the Flip bifurcation is indicated with the filled triangles; the other symbols are the same as those adopted in Figures 6 and 7.

Figure 11. Bifurcation diagram. The inlet temperatureTin, and the switchtime t′ (τ/n) are chosen as the bifurcation parameters. The solid curverepresents the locus of saddle-node bifurcation S2, and the dashed curverepresents the locus of saddle-node bifurcation S2′; the dashed-dotted lineis the asymptotic behavior for the unforced network.


follows. Equation 12 is based on the adiabatic assumption thatimplies that the reaction front moves at constant velocity alongthe reactor network, and also that this velocity is not dependenton the switch strategy. Conversely, under nonadiabatic condi-tions, the motion of the reaction front along the NTW issignificantly affected by the strategy, and the strategy influenceincreases as the feed temperature increases. In this regard, thespatio-temporal patterns are very illustrative. Figure 12 showsthe regime spatio-temporal patterns (over a switch timeτ) forstrategy 1 (Figure 12a) and for strategy 2 (Figure 12b) at aτ-value close to extinction conditions (τS2, τS′2

). For the sake ofclarity, Figure 12a is narrower, to illustrate the results on thesame time scale. In both cases, the fresh feed is entering in thefirst reactor (the leftmost panel in Figure 12a and b). The effectof the fresh feed on the temperature is visible in the lower leftcorner of both panels, as the combustion of VOCs happens atthe inlet NTW catalyst layers and does not enable the reactionfront to rise. However, after a short time interval (equal for boththe strategies), the reaction front forms. In the case of strategy1, the reaction front moves at a constant velocity (Vfr ≈ 1.25×10-4), crossing the entire first reactor in approximately a switchtime τ. The same happens in the first reactor for strategy 2, atthe very same velocity (Vfr ≈ 1.25× 10-4). In this latter case,however, during a switch timeτ, the front moves in the secondreactor as well. In this second reactor, the velocity of the reactionfront changes and increases up toVfr ≈ 1.625× 10-4. Hence,for strategy 2, we cannot identify a single reaction front velocity,and the averagedVfr is greater than that observed for strategy1. This fact explains the reduction of the factor of 2 observedin Figure 11.

The observed acceleration of the reaction front inside thesecond reactor can be easily motivated. Within the first reactor,the place where the reaction starts moves toward the dischargesection as time passes. Correspondingly, the fresh feed (atTin)progressively cools the first bed, proceeding from left to right.At times larger thant* (see Figure 13), the first reactor simplycools. Such a cooling under nonadiabatic conditions can leadthe gas temperature in the first reactor belowTin asTw < Tin.Of course, this implies that the second reactor is fed atT < Tin

for a substantial amount of time before switching. This is

confirmed by Figure 13. Consequently, as is well-known,34 thereaction front “accelerates” in the second reactor.

We conclude this section by characterizing theT-periodicisola at low switch time (τ) values. As previously mentioned,at low τ values, the dynamics is very complex for both switchstrategies.

For strategy 1, the isola at low switch times consists ofunstableT-periodic regimes, whereas for strategy 2, there is abranch of stableT-periodic regimes delimited by two Neimark-Sacker bifurcations.

A spatio-temporal pattern of theT-periodic regime betweenthe two Neimark-Sacker bifurcations is reported in Figure 14.As it can be observed, the spatio-temporal pattern is notcharacterized by continuous hot regions that are traveling alongthe NTW. The length and the maximum temperature of thethermal waves are, indeed, strongly variable along the NTWand thus the spatio-temporal patterns are not represented bycontinuous bands as for high switch time values. The reason isthat, in this range, the switch timeτ is small and, thus, thereaction, which is not able to develop a stable reaction front,occurs just in a small portion of the catalytic bed and for a shorttime interval.

Figure 12. Spatio-temporal patterns of the temperature in the gas phase of aT-periodic regime. The feed temperature in the gas phase isTin ) 240°C: (a)strategy 1 and a switch time ofτ ) 8000; and (b) strategy 2 and a switch time ofτ ) 14200. Each box corresponds to a reactor unit.

Figure 13. (s) Gas temperature at the exit of the first reactor for strategy2; (- - -) feed temperature in the gas phase. For both analyses,Tin ) 240°C, and the the switch time isτ ) 14200.


4. Comparison of Global PerformancesIn the case of VOC combustion considered here, some global

performances ofT-periodic traveling wave solutions are notaffected by the switch strategy. Figure 15 displays the maximumsolid- and gas-phase temperatures as the switch time is varied

for strategy 1 (Figure 15a) and strategy 2 (Figure 15b). A slightdependence of the maximum temperature on the switch time isobserved for strategy 1 (a slightly increase), whereas for thecase of strategy 2, the maximum temperature is almost constantthroughout the stability range. In any case, the maximum

Figure 14. Spatio-temporal gray-scale patterns of the temperature in the gas phase of aT-periodic regime for strategy 2 and a switch time ofτ ) 650. Thelevel of temperature is coded in shades of gray. Each box corresponds to a reactor unit. The feed temperature in the gas phase isTin ) 240 °C.

Figure 15. Maximum temperature in the NTW for different values of the switch timeτ: (a) strategy 1 and (b) strategy 2. The solid temperature is indicatedwith empty triangles (4), whereas the gas temperature is represented by open circles (O). The feed temperature in the gas phase isTin ) 60 °C.

Figure 16. Average outlet gas conversion during the period in the NTW for different values of the switch timeτ: (0) strategy 1 and (O) strategy 2. Thefeed temperature in the gas phase isTin ) 60 °C.


temperatures values attained for the two strategies are similar.To compare the two strategies, one must consider the perfor-mances attained at the same normalized switch time (that is,τfor strategy 1 corresponds to 2τ for strategy 2).

A similar strategy invariant behavior is observed for theconversion. Figure 16 shows the time-averaged gas conversionsfor both strategies versus the switch timeτ. When comparingthe predictions at the same normalized switch timeτ, it seemsthat the largest deviation is<0.5%. This implies that bothstrategies guarantee an almost complete combustion of theVOCs.

Conversely, the switch strategy strongly influences thetemperature profiles within the reactors. A simple descriptionof this fact can be obtained by comparing the average (bothtemporal and spatial) temperature as the switch time is varied,as reported in Figure 17.

It is apparent in Figure 17 that the average temperatures (solidand gas phases) for strategy 1 (see Figure 17a) are∼100 °C,which are greater (∼20%) than those predicted with strategy 2(see Figure 17b).

5. Conclusions

We analyzed and compared the stability ofT-periodic regimesof a network of three nonadiabatic catalytic fixed-bed reactorssubjected to two different periodic switch strategies: one inwhich, after one switch time, the first reactor of the sequenceis moved to the last place; and another, in which, after oneswitch timeτ, the last reactor of the sequence is moved to thefirst place.

Although the two strategies result in spatio-temporal sym-metries of the model equations, both isomorph to Z3×S1,different group generator transformations must be defined.Differences in symmetry properties reflect differences in thepredicted spatio-temporal patterns.

For both strategies, the switch time ranges ofT-periodicregimes are delimited by saddle-node bifurcations. Consistentlywith the theoretical predictions of Sheintuch and Nekhamkina,1

which were obtained under adiabatic conditions, the bifurcationswitch time values of strategy 2 are doubled, with respect tostrategy 1. This fact shows that the switch strategy can be usedto enlarge the narrow stability ranges of NTW.20 We found that,at low feed temperatures, this also is true for nonadiabaticconditions. Indeed at low feed temperature, the switch strategydoes not affect the reaction front velocity and the difference in

the switch time values are only linked to the difference in thereactor length covered by the reaction front for each switchstrategy. Whereas for higher feed temperatures, the theoreticalpredictions for the adiabatic system cannot be applied to thenonadiabatic case. Indeed, because of the external heat losses,the reaction front in strategy 2 moves faster, on average, withrespect to strategy 1.

The maximum temperature of the reaction front is not affectedby the switch strategy, whereas a slightly lower average con-version was observed for strategy 2, with respect to strategy 1.

Moreover, at higher feed temperature and for both strategies,an isola of symmetricT-periodic regimes is detected at lowswitch time values. This isola is constituted by unstableT-periodic regimes for strategy-1, whereas for the strategy-2 itis wider and it contains a branch of stableT-periodic regimesdelimited by two Neimark-Sacker bifurcations. For bothstrategies,T-periodic regimes at low switch time values arecharacterized by a strongly variable maximum temperature alongthe network (NTW).

Acknowledgment

Helpful suggestion by Dr. E. Vessicchio are warmly ac-knowledged. This work was partially funded by MIUR, andCNR (Gruppo Nazionale per la Difesa dai Rischi Chimico-Industriale ed Ecologici).

Nomenclature

a ) catalyst external surface areacp ) heat capacityC ) concentration of the species Adr ) reactor diameterD ) mass axial dispersion coefficientE ) activation energyf(t) ) forcing functionG ) linear operator defined in eq 5hfc ) heat-exchange coefficient between gas phase and catalysthfw ) heat-exchange coefficient between gas phase and reactor

wallk0 ) Arrhenius constantkg ) gas-phase axial heat conductivitykm ) mass-transfer coefficientks ) solid-phase axial heat conductivityI ) identity operator

Figure 17. Average temperature during the period in the NTW for different values of the switch timeτ: (a) strategy 1 and (b) strategy 2. The solidtemperature is represented by empty triangles (4), whereas the gas temperature is denoted by (O). The feed temperature in the gas phase isTin ) 60 °C.


L ) reactor lengthR ) gas constantr ) reaction rateS1) unit circlet* ) timet ) dimensionless timeT ) periodV ) gas flow rateV ) volume of the reactorz* ) axial coordinatez ) dimensionless axial coordinateZ3 ) cyclic group

Vectors and Matrices

F ) vector fieldx ) state vectorλ ) parameter vector

Greek Letters

η ) effectiveness factorΓ ) generic Lie group of spatial transformationγ ) generic spatial transformation of a Lie group∆ ) generic Lie group of temporal transformation∆H ) heat of reactionδ ) generic temporal transformation of a Lie groupε ) reactor void fractionεs ) catalyst porosityF ) densityτ ) dimensionless switch time

Subscripts and Superscripts

c ) refers to heat exchangerg ) gas phaseh ) energy balancein ) feedm ) massout ) outlet from the systems ) solid phasew ) wall

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ReceiVed for reView January 19, 2007ReVised manuscript receiVed June 11, 2007

AcceptedJune 11, 2007

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