Loop Reactor Design and Control for Reversible Exothermic Reactions

May 4, 2009 - show that the system exhibits the slow-switching asymptote and the complex ... many-units loop reactor in which a single exothermic reac...
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Ind. Eng. Chem. Res. 2009, 48, 5185–5192

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Loop Reactor Design and Control for Reversible Exothermic Reactions Roman Sheinman and Moshe Sheintuch* Department of Chemical Engineering, Technion-I.I.T., Technion City, Haifa 32 000, Israel

The loop reactor, in the form composed of several units, with feed and exit ports switching, is investigated for a reversible exothermic reaction like the synthesis of methanol. Unlike applications of VOC combustion, which have been extensively investigated and experimentally tested, the maximal temperature here is limited by equilibrium conditions and the advantage gained is due to the effectively periodic boundary conditions that describe the system: due to the rotating nature of the system, the stream undergoes cooling as it leaves the system, which in turn yields increasing conversions at an extent that depends on the parameters. We show that the system exhibits the slow-switching asymptote and the complex many-domains pattern that was identified for the irreversible case. The dynamic features within these domains are analyzed. Conversions are comparable to those in the commercially applied reactor with interstage cooling. Control procedures are suggested and tested. 1. Introduction The loop reactor (LR), in the form composed of several units, with switching of feed and exit ports between them (Figure 1), is one of the suggested technological solutions for lowconcentration VOC combustion. This is one of the conceptual solutions that combines a packed bed with enthalpy recuperation like the reverse-flow reactor1-3 (RFR). The RFR and the LR has been claimed to be advantageous for VOC combustion, due the high temperature rise achieved, as well as for reversible exothermic reaction, thanks to the spontaneous decliningtemperature zone that develops within the reactor. The purpose of this work is to analyze the LR for the latter application. The loop or ring reactor has been proposed by Matros4 and was simulated for a case of two units by Haynes and Caram5 or for three units by Barresi’s group6,7 for the applications described above. Sheintuch and Nekhamkina8-10 have generalized these results by studying the asymptotic solutions of a many-units loop reactor in which a single exothermic reaction occurs, showing the existence of a slow switching solution in which the forcing switching velocity (Vsw ) ∆L/τsw, i.e., unit length divided by switching time) and the spontaneous front velocity (Vfr) are synchronized and the pattern is “frozen” in moving coordinates. In that asymptote, the system is analogous to a loop-shaped bed rotating in the direction opposite to the flow at the speed of pulse propagation. The feed position is fixed in time. Another asymptote, that of a fast switching solution in which Vsw significantly exceeds Vfr, was also shown to exist. Moreover, between these two asymptotes we recently showed (Nekhamkina and Sheintuch11) the existence of many “finger” like domains of complex frequency-locked solutions that allow to extend significantly the operation condition domain, rendering the LR scheme more attractive for practical implementation. Comparison of the LR operation, within the slow-switching domain, with the RFR12,13 led to the conclusion that its main advantages are the significant reduction of the washout effect, the autothermal behavior, and the nearly uniform catalyst utilization. The main drawback of the LR is the very narrow window of switching velocities that sustain the solution in the frozen-patterns “finger” (see below). If control is incorporated, this domain is quite wide, as we show below. The first experimental implementations of the LR were recently reported for an almost isothermal process (selective catalytic reduction,

NH3 + NO, Fissore et al.14) and for the VOC combustion concept (ethylene oxidation, Madai and Sheintuch15). The fast switching operation domain (Vsw > Vfr), for a threeunit LR, was reported in a series of numerical studies;6-8,10 nonlinear dynamics and bifurcation analysis were conducted by Russo et al.16-18 and Altimari et al.19 A brief analysis of complex nonfrozen solutions in LR with different N was conducted in our previous works.8,9 To understand the advantage gained in the LR for VOC combustion, recall the effect of front velocity (Vfr) on the maximal temperature (Tm) in a fixed bed reactor. For the case of a single irreversible reaction in an ideal front (i.e., in an infinitely long system) Tm and Vfr admit the following relations (Wicke and Vortmeyer20): ∆Tm ) Tm - T0 ) ∆Tad Vfr ) VS

VS - Vfr , VS - LeVfr

∆Tad - ∆Tm Le∆Tad - Le∆Tm

(1)

The peak temperature of a front (in a once-through catalytic reactor) or of the rotating pulse (in a loop reactor) can be increased significantly by pushing the front by increasing the flow rate (Vs). Obviously, the front velocity cannot exceed the thermal front velocity Vfr ) Vsw ) Vs/Le which corresponds to an infinitely large temperature rise in the infinitely long system. In a finite length once-through reactor, the pulse will eventually leave the system and the extinguished state will prevail. Thus, some form of matched boundary conditions or external periodic forcing is required to sustain the periodic motion.

Figure 1. Representation of loop reactor operation, as a train of periodically switched fixed beds (left) and as a rotating bed (right).

10.1021/ie801333w CCC: $40.75  2009 American Chemical Society Published on Web 05/04/2009

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In reversible reactions, on the other hand, the maximal temperature is limited by equilibrium conditions. The advantage gained here is due to the effectively periodic boundary conditions that describe the system: the exit bed temperature is quite close to the feed temperature and, consequently, the stream undergoes cooling as it leaves the system. This cooling yields increasing conversions at an extent that depends on the spatial rate of cooling, the flow rate, and the reaction kinetics. The aim of the current work is to extend the information obtained for LR design for irreversible reactions, i.e., the two asymptotes and the qualitative structure of the operating domain, to designs for reversible reactions and use this understanding to assess the advantages and shortcoming of such design. The main alternative approaches for reversible exothermic reaction are the interstage cooling or the counter-current autothermal designs, which require external or internal heat exchangers, respectively. The structure of this work is the following: in the next section the model and its asymptotes are outlined, while their solutions are described in section 3 along with several control schemes. We use approximate kinetics, reduced from the detailed one as described in the Appendix. 2. Reactor Model

rA

r-A

(-∆H) ) 9.071 × 104

set I

∂T ∂2T ∂T + (FCP)f u ) ke 2 + (-∆H)(1 - ε)r(C, T) ∂t ∂z ∂z (2) ∂C ∂2C ∂C +u ) εDf 2 - (1 - ε)r(C, T) ∂t ∂z ∂z

r(CA, CB, T) ) k1(T)C - k2(T)(C0 - C); -Ea,i ki(T) ) Ai exp RT

( )

(3)

(4)

The rate equation has been reduced to a single-reactant presentation under the reasonable assumption that the massdispersion coefficients of the key reactant and product are identical and that a sufficiently long time has elapsed since the perturbation; under these conditions the two mass balances can

set II

exothermicity, B 16 5.5 input temperature (T ) 100 °C), θin - 8.2 - 2.8 input temperature (T ) 20 °C), θin - 15 - 5.0 Peclet number for heat conduction, PeT 413 413 390 390 Peclet number for mass diffusion, PeC Damkohler number, Da 0.017 11 Lewis number, Le 29 29 activation energy, γ 39 13 reaction enthalpy to activation energy, µ 1.6 2.9 8 ratio between direct and reverse reaction constants, ψ 3.2 × 10 9.1 × 1010 reactor length, L 1/2 1/2

be combined to eliminate the nonlinear reaction term and to find that CCH3OH ) C0 - C. In dimensionless form the model takes the form Le

1 ∂2θ ∂θ ∂θ + VS ) + Brˆ(x, θ); ∂τ ∂ζ PeT ∂ζ2 1 ∂2x ∂x ∂x + VS ) + rˆ(x, θ) ∂τ ∂ζ PeC ∂ζ2

(5)

where the reaction term is

( θ θγ+ γ )[1 - x{1 + ψ exp( (1θ-+µ)γγ )}] 2

(6)

µ)

E2 -∆H ) + 1; E1 E1

ψ)

A2 A1

The boundary conditions are those of once-through operation, i.e., the Danckwerts BC, which are applied at the feed and exit port: ζ ) ζin,

J mol

We shall assume that the feed includes 6.5% M of carbon monoxide in hydrogen. To simplify modeling, we assume that both direct and reverse reactions are of first order with Arrhenius dependence in temperature. The transport coefficients (ke, Df) and thermodynamic parameters (F, cp, and -∆H) are assumed to be constant. Under these assumptions the enthalpy and mass balances in a one-dimensional homogeneous system may be written for each unit in the following form: (FCP)eff

parameter

rˆ(x, θ) ) Da exp

In the simulations below we use an exothermic first-order reversible, activated reaction. We use the kinetic and thermodynamic parameters of the synthesis of methanol from CO and hydrogen with the latter being in excess. Since the feed typically includes CO2 and water, this reaction is one of the three that usually occur during catalytic methanol synthesis, and this was the problem considered numerically by Velardi et al.6 To gain some understanding, we assume only two reactants in the feed and that only one reaction occurs. CO + 2H2 {\} CH3OH

Table 1. Summary of Dimensionless Parameters

1 ∂θ ) V(θ - θin), Peθ ∂ζ

1 ∂x ) V(x - xin); Pex ∂ζ ∂x ∂θ ) )0 ζ ) ζout, ∂ζ ∂ζ

(7)

while continuity is applied between the units; i.e., the system is modeled as a single continuous reactor. Here conventional notation is used: tu0 T - T0 Tin - T0 , θ)γ , θin ) γ , z0 T0 T0 u L˜ E C , x ) 1 - , VS ) , L ) , γ ) C0 u0 z0 RT0 Az0 (FCp)e (-∆H)C0γ , Da ) exp(-γ), Le ) , B) (FCp)fT0 u0 (FCp)f (FCp)fz0u0 (FCp)fz0u0 , PeC ) (8) PeT ) ke Df

ζ)

z , z0

τ)

Note that we use arbitrary values (u0, z0) for the fluid velocity and the length scales in order to ensure that the corresponding dimensionless values (L, V) can be varied as independent parameters. We choose u0 ) 1 m/s, z0 ) 1 m, and report sometimes the dimensional reactor length and feed flow rate. Typically, in our simulations, as well as in practical situations, the ratio of solid to fluid heat capacities is large, Le . 1 (for this high-pressure case, Le ) 28.5) and PeT, PeC . 1. The parameters of the system were adopted from Velardi et al.6 and converted to a first-order presentation within a limited temperature range (see Appendix). The reference temperature T0 was chosen to be 200 °C.

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Figure 2. Bifurcation map showing domains of active solutions, in the plane of fluid vs switching velocity, for feed temperature of Tin ) 100 °C (left) or 20 °C (right). Both maps calculated for set I of parameters.

The many-port slow-switching asymptote was derived by Nekhamkina and Sheintuch9 and was shown to converge to the full model results as the number of ports is increased. In the slow-switching domain, the system can obtain a stationary solution in a moving coordinate system, ξ ) ζ - Vswτ, with its origin at the continuously moving feed position. The system at the steady moving state is described by 1 ∂2θ ∂θ + B · rˆ(x, θ) ) 0 - (Vs - VswLe) PeT ∂2ξ ∂ξ ∂x 1 ∂2x + rˆ(x, θ) ) 0 - (Vs - Vsw) PeCF ∂ξ2 ∂ξ

(9)

with rˆ(x, θ) defined in eq 6 and subject to the following boundary conditions: 1 ∂x at ξ ) 0: Pe ∂ξ ) Vs(x - xin); C

(| | )

1 ∂θ PeT ∂ξ at ξ ) ξL:

∂x ∂ξ

|

ξ)ξL

-

ζ)0

∂θ ∂ξ

ζ)ζL

) VS(θ0 - θin)

) 0, θx)1 ) θx)0

(10)

3. Results Following our previous study of irreversible reactions, we draw domains of unextinguished solutions in the plane of fluid velocity vs switching velocity. Figure 2 shows a grayscale plot of the possible operating windows where the conversion is not nil. Throughout this plane there exists another extinguished solution, and in the white domains it is the only solution. The maximal temperature is expected to fall approximately at Vsw ) Vs/Le, since the apparent convection term (eq 9) vanishes. As expected the first domain (or finger or “tongue”) falls around that value. Below, we elaborate further on the first domain and the control required for safe operation. The other domains fall around integer multiples of that value, i.e., Vsw ) nVS/Le, n ) 1, 2,... (the seven lines drawn to guide the eye fall at VS ) VswLe/ n). For large n the domains are indistinguishable. At lower feed temperature the same domain structure is still valid, but the domains are narrower. Especially, the even n domains are very narrow and may not be continuous. The first domain yields simple periodic solutions with small temporal variation in temperature, and in the limit of many reactors while keeping the same length, it yields an asymptotic solution that moves at constant shape and speed (see below). Figure 3 presents the temperature or conversion spatial profiles, as well as their phase plane presentations, for several switching velocities within the slow-switching domain at u ) 1 m/s. It

shows again that an ignited solution exists only for a limited range of switching velocities (for fixed Vs), while outside this domain the solution is extinguished. This domain exist around Vsw ) u/Leu0 ∼ 0.032. The largest temperature is limited by equilibrium conditions, independent of the switching velocity. The conversion jumps in two steps: at the entrance, due to the high temperature generated there, and toward the exit due to the cooling effect. The first jump is equilibrium-limited while the second is limited by kinetics and the spatial temperature gradient. Decreasing flow rate will increase the conversion in the second jump. This is unlike the case of irreversible reactions where the temperature increases with feed flow rate. Increasing switching velocity will lead somewhat higher conversions near the right boundary of that domain. When plotting this data in the phase plane we see that the second step (declining temperature) follows the equilibrium line until the temperature is too cold to sustain a reaction (Figure 3c). Note that there is no qualitative differences between the two sets which differ significantly in their activation energies. To avoid extinction, we should follow a constant u/Vsw ratio. Plots of conversion along these lines, within the fingers in Figure 2a, show that the state is ignited almost for the whole domain and the conversion declines with u (Figure 4) as expected from the analysis of the second conversion jump above. Thus, while a higher conversion can be achieved at lower flow rates in the intermediate- and fast-switching domains, the throughput achieved in the slow-switching domain is the largest, and the decision in which domain to operate should be based on the ease of control, as well as on the economics of reaction and separation. Let us discuss the dynamic within the various fingers (Figure 5): the behavior is simple-periodic within the slow-switching finger, showing one peak per cycle of the exit conversion and a simple one-peak spatial profile. Fourier spectrum of exit conversion exhibits the dominant frequency, which is the forcing frequency, and its harmonics (Figure 6, column 1). This is also evident from the spatiotemporal pattern (Figure 5a). As we move to the other fingers, the temperature profile exhibits a complex profile with several peaks (hot zones) within the system, the exit conversion is multipeak or even aperiodic and the spectrum includes more frequencies. At very high switching velocities the system approaches the corresponding asymptote showing a dominant frequency that is independent of Vsw (Figure 6). This is also evident from the spatiotemporal pattern (Figure 5c). Numerical Solution of Quasi-Steady-State Problem. Comparison of the asymptotic model with simulation results of 32 units loop reactor of same total length (Figure 7) affirms again the validity of the many-unit asymptote (see ref 9 for detailed comparison). The asymptotic steady-state solution of eqs 9 and

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Figure 3. Temperature (a) and conversion (b) profiles along the reactor for the slow-switching velocity solution, and their phase plane representation (c), for set I (upper row) and set II (lower row). Tin ) 100 °C for both sets.

Figure 4. Time-average conversion and production rate, along lines with constant VS/Vsw ratio (set I) shown in Figure 2a.

Figure 5. Examples of simple-periodic behavior within the slow-switching finger (a) and of more complex behaviors of frequency locked states (b,c), showing the temperature in the time vs space plane. (a) Conditions are of the first slow-switching finger (Vsw ) 0.140 m/s; u ) 4.141 m/s), (b) of finger number 4 (Vsw ) 0.140 m/s; u ) 1.236 m/s), and (c) of finger number 7 (Vsw ) 0.140 m/s; u ) 0.468 m/s).

10 was obtained by a 200-node finite-difference approach using Mathematica. Control Design. The main drawback of the loop reactor is the narrow domain of switching velocities that support

traveling pulses; the slow-switching domain becomes narrower with decreasing adiabatic temperature rise (∆Tad or B) and forms a U- or V-shaped domain in the (B,Vsw) plane11 shown in Figure 8. Since not all parameters (especially Le)

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Figure 6. Comparison of dynamic behavior of LR at different operating regimes showing the temperature profile, the exit conversion temporal behavior, and the Fourier transform of the signal, (set I, Tin ) 100 °C). For all tests Vsw ) 0.140 m/s; u ) 4.1413, 1.7983, 1.4971, 1.2363, 0.952 16, 0.724 05, 0.468 27 m/s, respectively, for seven tests.

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are known exactly, a control mechanism was simulated in several numerical studies and was applied in the experimental study. Barresi et al. (2002) suggested to use the switching velocity as an actuator and simulated the behavior with the following feedback control: For two consecutive reactors the feed is switched from one reactor to the following when the temperature measured at the entrance of the catalytic area in the second reactor reaches Tsw2 (a threshold temperature) and the temperature at the entrance of the first reactor past the feed drops below Tsw1. In practice it is the former temperature (Tsw2) that triggered the switch. The stability of a rotating pulse with a switching velocity controlled to minimize the deviation of a reading T(z*), measured at certain z*, from a certain set point Ts was recently studied in our group.8 The approaches described above require that two parameters be determined (z*,Ts) or (z*,Tsw1) for proper design. Here we want to design a control that identifies the pulse peak and determines its velocity and tries to keep it at a constant distance (on average) from the feed, thus reducing the number of control parameters to one (z*, or the dimensionless ζ*). To identify the pulse peak, we note that we should see a periodic signal from each of the thermocouples placed in the reactor. These signals are masked by some noise. To filter out the effect of noise and determine the pulse velocity we propose the following steps: 1. The controller compares its temporal reading, within a certain time interval, with a prespecified function, usually a parabola. Once the peak is identified the program reports that the hot spot has passed the sensor. 2. From knowledge of the temporal interval of peak passage between adjacent sensors, the average velocity is calculated. 3. Using the present peak position and the average velocity one of the following two control laws has been applied: Proportional Control. The deviation of the estimated front position from a certain set point is used to adjust Vsw according to

Vsw ) KP(ζpeak - ζ*)

(11)

Recall that Vsw ∼ O(1/Le) and ζ* ∼ O(1) so Kp ∼ O(1/Le). Proportional-Integral (PI) Control. The switch velocity is adjusted every switch like Vsw,i+1 ) Vsw,i+1 + K1(ζ*i - ζsp) + K2(ζ*i - ζ*i-1) (12) where ζi is the peak position at the reactor. The drawback of the proposed estimation algorithm is the measurement time lag. The delay is induced because the measurement is updated only when peak leaves the measurement window. Therefore, the time delay is equal to the window’s width. Reducing the window width will reduce the delay, paying by reduced precision of estimation. The plot of operating domain is reconstructed now under control in the plane of B vs Vsw: now Vsw is not fixed, but rather the result of control. After pseudoperiodic conditions are obtained the corresponding switch velocity was recorded and plotted (Figure 8). Following an experiment of decreasing feed concentration (or B), using the last point as initial conditions for a new set, the locus of such pseudoperiodic states was recorded until extinction occurred (Figure 8). The extinction point in the closed-loop system is quite close to that simulated for uncontrolled system (open loop) system for proper choice of sensor location. The control is effective: the lowest ∆Tad that sustains a reaction is not strongly dependent on the sensor position or threshold temperature. Forcomparisonweshowasimilarfigureusingproportional-integral control of irreversible reaction (using a set of parameters in ref 8). We analyzed the behavior of a loop reactor with a reversible reaction. The reactor exhibits similar behavior to the well-studied case of irreversible reaction, like the slowswitching domain of “frozen” solution and multiple-switching

Figure 7. The asymptotic solution, compared to 32 reactors in loop model (set I). Vsw ) 1/Le, VS ) 1.

Figure 8. Testing the proportional (a) or proportional-integral control law (b): periodic ignited solutions obtained by decreasing B values are denoted in the (B, Vsw) plane (set I); (c) shows the proportional-integral control law test on irreversible first-order reaction.

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domains of complex solutions (Figure 1). The domains have a classical form of “tongues” that fall around constant values of LeVsw/VS. Unlike the irreversible reactions, the temperature and conversion of reversible reaction within the slow-switching domain are limited by equilibrium conditions, which are independent of the switching velocity. The conversion increases in two steps (Figure 2): at the entrance, due to the temperature rise, and is limited by equilibrium. The second jump happens toward the exit, due to the cooling effect, and is limited by kinetics and the spatial temperature gradient. Decreasing flow rate will increase the conversion in the second jump. This is unlike in the case of irreversible reactions where the temperature increases with feed flow rate. Increasing switching velocity will lead to somewhat higher conversions near the right boundary of the slow-switching domain (Figure 3). As we move to the other domains (tongues), the temperature profile becomes more complex featuring several peaks (hot zones) within the system. The exit conversion is multipeak or even aperiodic, which is illustrated using frequency content of Fourier spectrum (Figure 5). At very high switching velocities the system approaches the corresponding asymptote showing a dominant frequency that is independent of switch velocity. Appendix: Rate Expressions and Parameter Estimation The enthalpies and other physical properties are estimated using Unisim, Peng-Robinson property prediction package.21 Another method, the Zudkevich-Joffee property prediction package, is the modification of Peng-Robinson method and is designed to handle hydrogen-containing mixtures. The difference between the methods is less than 0.1%. The catalyst properties and bed dimensions are those used by Velardi et al.6 The original rate equations for methanol synthesis on a ZnO/ CuO/Al2O3 catalyst were suggested by Graaf et al.22 Here we approximate them by a linear expression assuming hydrogen to be in excess. rA )

kps,A ′ KCO pCO pH2 2 (1 + KCO pCO + KCO2 pCO2)[pH2 + (KH2O /KH1/22 )pH2O pH1/22 ] -1 mol · s-1 · kgcatalyst (A1)

pCH3OH r-A ) rA Kp,A pH2 2 pCO

Table A1. Parameters Used in Simulations total length, L void fraction, ε catalyst density, FS catalyst specific heat, CP,S catalyst thermal conductivity, λS pellet diameter, dp total pressure, P superficial inlet flow rate, Fin feed temperature av mol wt of feed, MW feed density, FF feed specific heat, CP,F feed thermal conductivity, λF feed diffusivity, Dmx feed viscosity, µ

KCO ) 7.99 × 10-7e58100/(RT) bar-1

KCO2 ) 1.02 × 10-7e67400/(RT) bar-1 Kp,A ) 105139/T-12.621 ) 2.3933 × 10-13e98385/(RT) bar-2 KH2O /KH1/22 ) 4.13 × 10-11e104500/(RT) bar-1/2 (A3) We approximated the rate of the forward reaction as a first order with respect to CO in the temperature range of 300-1200 K and re-estimate the activation energy and preexponent factor by fitting the data for the forward reaction. Parameters of reverse reaction are estimated from ratio of reverse to direct reactions: Set I. We assume that the denominator varies slowly with conversion, so that it is mainly temperature dependent. For a

0.5 0.4 1750 1000 0.33 0.0054 5 32.65 100 or 20 4.02 6.2593 7.3535 × 103 0.1917 0.0054 1.2964 × 10-4

m kg · m-3 J · kg-1 · K-1 W · m-1 · K-1 m MPa mol · m-2 · s-1 °C g · mol-1 kg · m-3 J · kg-1 · K-1 W · m-1 · K-1 m2 · s-1 Pa · s

Table A2. Kinetic Data parameter

set I

set II

reaction enthalpy (-∆H), J/mol pre-exponential constant kest, mol · s-1 · kgcatalyst-1 activation energy Ea,1, J/mol activation energy, back reaction Ea,-1, J/mol

98385 7.379 × 1012

90710 9.083 × 1010

153547 251932

51800 142510

set of concentrations that represent feed conditions (at P ) 25 bar and pi ) Pyi) yCO ) 0.045; yCO2 ) 0.018; yCH3OH ) 0.0065; yH2O ) 0.002; yH2 ) 0.92 we plotted the term rA ) pCO

k′psAKCO pH3/22

(

(1 + KCO pCO + KCO2 pCO2) pH1/22 +

KH2O KH1/22

pH2O

)

(A4)

vs 1/T and determined the apparent activation energy within the temperature range of 300-1200 K. This yielded the set of following parameters: kest ) 7.379 × 1012e-153547/(RT) mol · s-1 · kgcatalyst-1. Note the high activation energy is due to the temperature effect on adsorption equilibrium coefficients, especially on KH2O (actual activation energy is due to k′psA and KCO; see Table A2). The reaction enthalpy was set at Ea,1 Ea,-1 ) 98 385 J/mol, which is quite close to theoretical value of reaction enthalpy 91000 J/mol. Set II. Assuming that temperature is high so that the terms KH2OpH2O , 1 and KCO2pCO2 , 1, the rate is rA )

(A2)

′ -1 kps,A ) 2.69 × 107e-109900/(RT)mol · s-1 · kgcatalyst

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kps,A ′ KCO pCO pH2 (1 + KCO pCO)

(A6)

and it varies from zero order with apparent activation energy of 109 000 J/mol, under feed conditions, to a first order with apparent activation energy of 51 800 J/mol at sufficiently high temperatures when KCOPCO , 1. We take the latter case since it applies for most cases studied here where the catalyst is operating at T > 600 K (e.g., Figure 4). The kinetic data is presented in Table A2. Acknowledgment Work supported by the US-Israel Binational Science Foundation. Moshe Sheintuch is a member of the Minerva Center of Nonlinear Dynamics. Literature Cited (1) Matros, Y. Sh.; Bunimovich, G. A. Reverse-Flow Operation in Fixed Bed Catalytic Reactors. Catal. ReV. 1996, 38 (1), 1–68.

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(15) Amadi, A.; Sheintuch, M. Experimental Verification of Loop Reactor Performance: Ethylene combustion. AIChE J. 2008,54(9),2413– 2422. (16) Russo, L.; Continillo, G.; Crescitelli, S. Computation of frequency locking regions for a discontinuous periodically forced reactor. Comput. Chem. Eng. 2004, 28 (1-2), 187–194. (17) Russo, L.; Mancusi, E.; Maffettone, P. L.; Crescitelli, S. Non-linear analysis of a network of 3 continuous stirred tank reactors with periodic feed switching: symmetry and symmetry-breaking. Int. J. Bifurcation Chaos Appl. Sci. Eng. 2004, 14, 1325–1341. (18) Russo, L.; Altimari, P.; Mancusi, E.; Maffettone, P. L.; Crescitelli, S. Complex dynamics and spatio-temporal patterns in a network of distributed chemical reactors with periodic feed switching. Chaos Solitons Fractals 2006, 26, 682–706. (19) Altimari, P.; Maffettone, P. L.; Crescitelli, S.; Russo, L.; Mancusi, E. Nonlinear dynamics of a VOC combustion loop reactor. AIChE J. 2006, 52 (8), 2812–2822. (20) Wicke, E.; Vortmeyer, D. Zu¨ndzonen heterogener Reaktionen in gasdurchstro¨mten Ko¨rnerschichten. Ber. Bunsenges. 1959, 63, 145–152. (21) Graaf, G. H.; Stamhuis, E. J.; Beenackers, A. A. C. M. Kinetics of the low-pressure methanol synthesis. Chem. Eng. Sci. 1988, 43, 3185–3195. (22) Unisim Design, R370, 2005, Honeywell.

ReceiVed for reView September 3, 2008 ReVised manuscript receiVed February 22, 2009 Accepted March 3, 2009 IE801333W